arXiv:2006.08578v2 [math.NT] 2 Jul 2021 oe oyoil ahe nain,Sde product. Sudler (2020): invariant, Kashaev polynomial, Jones ergre stesaeieobto hre atce typical A particle. charged the a is of invariant whe orbit an physics, spacetime the quantum as theoretical regarded in be arise invariants knot Quantum Introduction 1 nt,wihi ie by given is which knot), 1 hogotteppr mt useul0 n mt products empty and 0, equal sums empty paper, the Throughout Keywords: unu nainso yeblckosand knots hyperbolic of invariants Quantum on yBti n rpeui h aeo ag ata quot partial large of case the in of value behavior Drappeau the relate and limiting Bettin universal by the found and from irrational, deviates quadratic a asymptotics to convergents fraction tinued rssfo unu nain ftefiueegtko,and knot, figure-eight the find of We invariant quantum product. trigonometric a from arises oaqeto fLubinsky. of produ Sudler question such a for to bounds lower and upper asymptotic lish xrm auso rgnmti products trigonometric of values extreme 17,5K6 10,26D05 11L03, 57K16, 11J70, nti ae esuyterlto ewe h function the between relation the study we paper this In Email: J 4 1 otne rcin udai rainl unu oua om c form, modular quantum irrational, quadratic fraction, continued ,n hitp Aistleitner Christoph ( [email protected] n q ea tnd ta1–5 03Bdps,Hungary Budapest, 1053 13–15, Re´altanoda utca clrdJnsplnma fteknot the of polynomial Jones -colored = ) J nttt fAayi n ubrTheory Number and Analysis of Institute N 4 X 2 1 1 ∞ lre ´niIsiueo Mathematics Alfr´ed of R´enyi Institute =0 , tyegse3,81 rz Austria Graz, 8010 30, Steyrergasse 0 1 ota fSde’ rgnmti rdc,adestab- and product, trigonometric Sudler’s of that to q rzUiest fTechnology of University Graz − nN Y j N =1 Abstract (1 J 4 − 1 , 1 0 q 1 n pt osatfco ln con- along factor constant a to up n ec Borda Bence and − j )(1 and ahmtc ujc Classification Subject Mathematics − [email protected] q n + j ) n , K qa 1. equal eso htits that show we 4 = hthsbeen has that t nresponse in cts 1 , ≥ 1 2 J tefigure-eight (the 4 xml fsuch of example 2 et.We ients. 1 , , 0 eako can knot a re Sudler’s which , olored defined for roots of unity q. For a fixed q, the mapping n J41,n(q) is periodic in n, and so the definition can be extrapolated backwards in7→n to give
∞ J (q)= (1 q)(1 q2) (1 qN ) 2 (1) 41,0 | − − ··· − | N=0 X for a root of unity q. Note that both sums actually have only finitely many terms for any root of unity q. The figure-eight knot is the simplest hyperbolic knot; for other hyperbolic knots K one obtains formulas for JK,0 which are of a somewhat similar but more complicated nature. The so-called Kashaev invariant K = J (e(1/n)), n = 1, 2,... is another quantum invariant of the knot K; h in K,n here and for the rest of the paper e(x) = e2πix. The Kashaev invariant plays a key role in the volume conjecture, an open problem in knot theory which relates quantum invariants of knots with the hyperbolic geometry of knot complements. For more general background information, see [16]; in the context of our present paper we refer to [6] and the references therein. The functions JK,0 also have an interpretation as quantum modular forms as introduced by Zagier [18], and are predicted by Zagier’s modularity conjecture to satisfy an approximate modularity property. For the figure-eight knot the modu- larity conjecture has been established [6, 9]; that is, the function J41,0(q) satisfies a remarkable modularity relation of the form J (e(γr))/J (e(r)) ϕ (r), where 41,0 41,0 ∼ γ γ SL2(Z) acts on rational numbers r in a natural way, and the asymptotics holds∈ as r along rational numbers with bounded denominators. The ratio → ∞ J41,0(e(γr))/J41,0(e(r)) as a function of r in general has a jump at every rational point, and consequently the asymptotics of J41,0(e(a/b)) along rationals a/b with b is quite involved. It is known [2] that →∞ 3/2 n Vol(41) J4 ,0(e(1/n)) exp n as n , 1 ∼ √4 3 2π →∞ where 5/6 Vol(4 )=4π log (2 sin(πx)) dx 2.02988 1 ≈ Z0 is the hyperbolic volume of the complement of the figure-eight knot; this follows from the fact that the volume conjecture, as well as its stronger form, the arith- meticity conjecture have been verified for 41. Bettin and Drappeau [6, Theorem
3] found the asymptotics of J41,0(e(a/b)) for more general rationals a/b in terms of their continued fraction expansions: if a/b = [a0; a1,...,ak], ak > 1, is a sequence of rational numbers such that (a + + a )/k , then 1 ··· k →∞ Vol(4 ) log J (e(a/b)) 1 (a + + a ) . (2) 41,0 ∼ 2π 1 ··· k 2 The result applies to a large class of rationals, including 1/n = [0; n], as well as to almost all reduced fractions with denominator at most n, as n . Verifying a →∞ conjecture made by Bettin and Drappeau, in this paper we will show that (2) in general fails to be true without the assumption (a1 + + ak)/k . The individual terms in (1) can be expressed in terms··· of the→∞ so-called Sudler products, which are defined as
N P (α) := 2 sin(πnα) , α R. (3) N | | ∈ n=1 Y This could also be written using q-Pochhammer symbols as
P (α)= (q; q) = (1 q)(1 q2) (1 qN ) with q = e(α), N | N | | − − ··· − | but (3) seems to be the more common notation. The history of such products goes back at least to work of Erd˝os and Szekeres [8] and Sudler [17] around 1960, and they seem to arise in many different contexts (see [15] for references). Bounds for such products also play a role in the solution of the Ten Martini Problem by Avila and Jitomirskaya [3]. It is somewhat surprising that, despite the obvious connection between (1) and (3), we have not found a reference where both objects appear together. A possible explanation is that (1) is only well-defined when q = e(a/b) with a/b being a rational, while the asymptotic order of (3) as N is only interesting when α is irrational. We will come back to this issue→ in ∞ Proposition 3 below. Note that for any rational number a/b we have PN (a/b) = 0 whenever N b; b 1 2 ≥ − in particular, this means that J41,0(e(a/b)) = N=0 PN (a/b) . The asymptotics (2) a fortiori holds for more general functionals of the sequence (PN (a/b))0 N
b 1 1/c − Vol(4 ) log P (a/b)c 1 (a + + a ) , (4) N ∼ 4π 1 ··· k N=0 ! X and also Vol(41) log max PN (a/b) (a1 + + ak) . (5) 0 N
3 (2), (4) and (5) give precise results for a large class of irrationals, but not when the sequence ak is bounded; our main result concerns this case. Recall that α is badly approximable if and only if ak is bounded, and that α is a quadratic irrational if and only if ak is eventually periodic; in particular, quadratic irrationals are badly approximable.
Theorem 1. Let α be a quadratic irrational. For any real c> 0 and any k 1, ≥ q 1 1/c k− c log PN (pk/qk) = Kc(α)k + O (max 1, 1/c ) ! { } NX=0 and log max PN (pk/qk)= K (α)k + O(1) 0 N 0. The implied constants depend only on α. ∞ In particular, we have
log J41,0(pk/qk)=2K2(α)k + O(1).
The proof of Theorem 1 is based on the self-similar structure of quadratic irra- tionals; that is, on the periodicity of the continued fraction. In fact, it is not difficult to construct a badly approximable α for which the result is not true. Quadratic irrationals exhibit a remarkable deviation from the universal behav- ior of irrationals with unbounded partial quotients. In contrast to (2), (4) and (5), the constants Kc(α) and K (α) in Theorem 1 are in general not equal to each ∞ other, or to Vol(41)a(α)/(4π); here a(α) = limk (a1 + + ak)/k denotes the →∞ average partial quotient, which is of course simply the average··· over a period in the continued fraction expansion. We have not been able to calculate the precise value of Kc(α) for any specific quadratic irrational and for any c; as far as we can say, this seems to be a difficult problem. The precise value of K (α) is known for ∞ some quadratic irrationals with very small partial quotients, as a consequence of results in [1]; for α with larger partial quotients, calculating K (α) precisely also ∞ seems to be difficult. However, it is possible to give fairly good general upper and lower bounds for Kc(α) and K (α) in terms of the partial quotients. Recall that for any quadratic ∞ irrational α, we have log qk = λ(α)k + O(1) with some constant λ(α) > 0; see Section 3 for a simple way of computing λ(α) from the continued fraction. We start with three simple observations. First, for any rational number a/b with
4 (a, b) = 1, the identity2
b 1 b 1 − − Pb 1(a/b)= 1 e(na/b) = 1 e(j/b) = b (6) − | − | | − | n=1 j=1 Y Y provides the trivial lower bound max0 N 0, we also have b 1 1/c b 1 1/c − − 1 c c PN (a/b) max PN (a/b) PN (a/b) , (8) b 0 N
Finally, we establish an antisymmetry of the sequence (PN (a/b))0 N Proposition 2. For any rational number a/b with (a, b)=1, and any integer 0 N < b, ≤ log PN (a/b) + log Pb N 1(a/b) = log b. − − Note that Proposition 2 immediately implies that log max PN (a/b) + log min PN (a/b) = log b, 0 Nfactorization (xb 1)/(x 1) = −1 → − − b (x e(j/b)). j=1 − Q 5 From (7) it follows that Kc(α) and K (α) exceed Vol(41)a(α)/(4π) when- ∞ ever the quadratic irrational α has relatively small partial quotients. For in- 1+√5 √ 1+√13 stance, 2 = [1; 1, 1, 1,... ], 2 = [1; 2, 2, 2,... ], 2 = [2; 3, 3, 3,... ], √ 1+√29 √ 5 = [2; 4, 4, 4,... ], 2 = [3; 5, 5, 5,... ] and 10 = [3; 6, 6, 6,... ] satisfy 1+ √5 1+ √5 K log 0.4812, K (√2) log(1 + √2) 0.8814, ∞ 2 ≥ 2 ≈ ∞ ≥ ≈ 1+ √13 3+ √13 K log 1.1948, K (√5) log(2 + √5) 1.4436, ∞ 2 ≥ 2 ≈ ∞ ≥ ≈ 1+ √29 5+ √29 K log 1.6472, K (√10) log(3 + √10) 1.8184, ∞ 2 ≥ 2 ≈ ∞ ≥ ≈ (11) whereas Vol(4 ) 1+ √5 Vol(4 ) 1 a 0.1615, 1 a(√2) 0.3231, 4π 2 ≈ 4π ≈ Vol(4 ) 1+ √13 Vol(4 ) 1 a 0.4846, 1 a(√5) 0.6461, 4π 2 ≈ 4π ≈ Vol(4 ) 1+ √29 Vol(4 ) 1 a 0.8077, 1 a(√10) 0.9692. 4π 2 ≈ 4π ≈ In particular, the sequence of convergents to these quadratic irrationals violate (2), (4) and (5), demonstrating that the condition (a + + a )/k cannot be 1 ··· k → ∞ removed. For a quadratic irrational α with large partial quotients, the constants Kc(α) and K (α) are nevertheless close to Vol(41)a(α)/(4π). Indeed, from results of ∞ Bettin and Drappeau [6, Theorem 2 and Lemma 15] it follows that for any rational a/b = [a0; a1,...,ak], ak > 1, Vol(4 ) log J (e(a/b)) = 1 (a + + a )+ O(A + k log A) (12) 41,0 2π 1 ··· k with A = 1 + max1 ℓ k aℓ and a universal implied constant. A fortiori, for any real ≤ ≤ c> 0, b 1 1/c − c Vol(41) log PN (a/b) = (a1 + +ak)+O(A+k max 1, 1/c log A) (13) ! 4π ··· { } NX=0 and Vol(41) log max PN (a/b)= (a1 + + ak)+ O(A + k log A). (14) 0 N 0, 1 1 K (α) + λ(α); c ≥ c 2 in light of (7) and (9), this is nontrivial for 0 Proposition 3. Let α = [a0; a1, a2,... ] be an irrational number with convergents p /q = [a ; a ,...,a ]. For any integers k 1 and 0 N < q , k k 0 1 k ≥ ≤ k log Ak log PN (α) log PN (pk/qk) | − |≪ ak+1 with Ak = 1 + max1 ℓ k aℓ and a universal implied constant. ≤ ≤ In particular, for any real c> 0, q 1 1/c q 1 1/c k− k− log A log P (α)c log P (p /q )c k N − N k k ≪ a N=0 ! N=0 ! k+1 X X and log Ak log max PN (α) log max PN (pk/qk) . 0 N 7 Applying the transfer principle to a quadratic irrational α, Theorem 1 can thus be restated in the irrational setting as 1/c M K (α) log P (α)c = c log M + O (max 1, 1/c ) , N λ(α) { } N=0 ! (16) X K (α) log max PN (α)= ∞ log M + O(1), 0 N M ≤ ≤ λ(α) where λ(α) > 0 is defined by log qk = λ(α)k + O(1), as before. For an arbitrary irrational α the reflection principle becomes, with the notation of Proposition 3, log Ak log PN (α) + log Pq N 1(α) = log qk + O ; k− − a k+1 in particular, log Ak log max PN (α) + log min PN (α) = log qk + O . (17) 0 N PN (α) lim inf PN (α) < and lim sup > 0 N ∞ N N →∞ →∞ for all irrational α. Note that the relations in the previous line also follow from the identity (6) and the transfer and reflection principles; in fact, the limsup relation is a far-reaching generalization of our trivial lower bound (7). More re- cently, Grepstad, Kaltenb¨ock and Neum¨uller [10] established the remarkable rela- 1+√5 tion lim inf PN ( 2 ) > 0, thus answering the question of Erd˝os and Szekeres in the negative. On the other hand, however, in the paper [11] by the same authors it was shown that for the special irrational α = [0; a, a, a, . . . ]=(√a2 +4 a)/2 one − has lim inf PN (α) = 0 whenever a is sufficiently large. Thus, rather remarkably, the question whether lim inf PN (α) > 0 or = 0 depends on the actual size of the partial quotients of α in a very sensitive way. Numerical experiments suggested 8 a similar change of behavior for large values of PN : for α = [0; a, a, a, . . . ] it was conjectured that lim sup P (α)/N < or = , depending on the size of a. This N ∞ ∞ problem was settled in [1], where it was proved that for α = [0; a, a, a, . . . ], PN (α) lim inf PN (α) > 0 and lim sup < if a 5 (18) N N N ∞ ≤ →∞ →∞ and PN (α) lim inf PN (α) = 0 and lim sup = if a 6. (19) N N N ∞ ≥ →∞ →∞ Regarding general badly approximable irrationals α, Lubinsky [15] proved that log P (α) log N; equivalently, | N |≪ c1 c2 N − P (α) N (20) ≪ N ≪ with some constants c1,c2 and implied constants depending on α. Let c1(α) resp. c2(α) denote the infimum of all c1 resp. c2 for which (20) holds; Lubinsky remarked that it is an interesting problem to determine these constants. The reflection principle (17) immediately shows that given a badly approximable α and a real c 1+c constant c, we have P (α) N − P (α) N , with implied constants N ≫ ⇔ N ≪ depending on α. Therefore c2(α)= c1(α) + 1, which is a striking general relation that seems not to have been noticed so far. Thus establishing the optimal value of c1 and that of c2 in (20) are actually one and the same problem. For a quadratic irrational α our main result (16) shows that c2(α)= c1(α)+1 = K (α)/λ(α), and in fact ∞ c1(α) PN (α) 0 < lim inf N PN (α) < and 0 < lim sup c (α) < . N ∞ N N 2 ∞ →∞ →∞ In particular, PN (α) lim inf PN (α) > 0 lim sup < K (α)= λ(α), N ⇐⇒ N N ∞ ⇐⇒ ∞ →∞ →∞ where the last condition means that our trivial lower bound (7) holds with equality. Note that this relation also explains why the behavior of the liminf and the limsup in (18) and (19) changes at the same critical value of a; this is also reflected in the six examples we gave in (11), where we actually have equality everywhere except for √10, in which case the inequality is strict. It seems to be a difficult problem to give a complete characterization of all quadratic irrationals α such that K (α)= λ(α); this is the subject of an upcoming paper of Grepstad, Neum¨uller ∞ and Zafeiropoulos [13]. 9 The results in our paper allow us to give a fairly precise estimate for the constants c (α),c (α) in the question of Lubinsky. From (15) with c = we 1 2 ∞ obtain that for any quadratic irrational α, Vol(4 ) a(α) log A(α) c (α)= c (α)+1= 1 + O 2 1 4π · λ(α) λ(α) with a universal implied constant. In particular, for α = [0; a, a, a, . . . ] we have Vol(41) a Vol(41) a c2(α)= c1(α)+1= + O(1) = + O(1). 4π · a+√a2+4 4π · log a log 2 The discussion above shows that K (α) K ( 1+√5 ) = log 1+√5 for all ∞ ∞ 2 2 ≥ 1+√5 quadratic irrationals. We do not know whether Kc(α) Kc( 2 ) for all quadratic irrationals and all c > 0; in fact, we do not even≥ know the precise value of 1+√5 Kc( 2 ) for any c. We mention, however, that numerical evidence found by Zagier [18] and Bettin and Drappeau [6] suggests that for the golden mean we 1+√5 have log J41,0(e(pk/qk)) 1.1k; that is, K2( 2 ) 0.55. In this context, it is an≈ interesting question to characterize≈ those values of N for which particularly large resp. small values of PN (α) occur. It is also interesting to estimate the relative number of indices which generate such values of PN (α). This would shed some light on the relation between the numbers Kc(α) and K (α) in M ∞ c Theorem 1 and (16). Essentially, the problem is whether the sum N=0 PN (α) is dominated by a very small number of indices N which produce particularly P large values of PN , or if there are enough such indices so that the full sum is of a significantly different asymptotic order than its maximal term. We plan to come back to all these questions in a future paper. Finally, we mention a further open problem. In [18] Zagier introduced the function h(x) = log (J41,0(e(x))/J41,0(e(1/x))). A conjecture of Zagier, established by Bettin and Drappeau in [6], implies that h has jumps at all rational points. Zagier also suggested that h(x) is continuous at irrational values of x (more pre- cisely, since h is formally only defined for rational x, the conjecture is that h can be extended to all reals such that it is continuous at irrational values). Let α be a quadratic irrational whose continued fraction expansion is of the simple form α = [0; a, a, a, . . . ], and let pk/qk be its convergents. Then it is easily seen that h(pk/qk) = log (J41,0(e(pk/qk))) log (J41,0(e(pk 1/qk 1))). Thus, while we cannot − − − prove that h(pk/qk) converges as k , as a consequence of Theorem 1 we can 1 K → ∞ at least conclude that K− h(p /q ) converges as K . Note that if k=1 k k → ∞ the error term in Theorem 1 could be reduced from O(1) to o(1), then we could P deduce that actually h(pk/qk) itself converges as k , so such a conclusion is just beyond the reach of our theorem. Finally, note→ ∞ that Theorem 1 implies 10 that if h can indeed be continuously extended to α, then the only possible value is h(α)=2K2(α) with K2(α) from Theorem 1. Thus our results can be seen as progress towards Zagier’s problem, while it seems that there is still a long way to go for a full solution of the problem. From the discussion above, one might expect that the problem requires different approaches according to whether the partial quotients of α are large (say, as in the case (a1 + + ak)/k ), or small (say, bounded). The most difficult case could be the one··· when the→∞ partial quotients of α are small, but there is no particular structure such as periodicity. 2 General rationals and irrationals Recalling (8), to deduce (4) and (5) from (2) we only need to show that the condition (a1 + + ak)/k implies that log b/(a1 + + ak) 0; indeed, this will show that··· for any c>→0, ∞ ··· → b 1 1/c − c log PN (a/b) log max PN (a/b). ∼ 0 N From the recursion satisfied by the convergents we get b (a1 + 1) (ak + 1). Letting a =(a + + a )/k, the AM–GM inequality gives≤ ··· k 1 ··· k log b log((a + 1) (a + 1)) log(a + 1) 1 ··· k k 0, a + + a ≤ a + + a ≤ a → 1 ··· k 1 ··· k k and we are done. To deduce (13) and (14) from (12), simply note that log b log(a +1)+ + log(a +1) = O(k log A), ≤ 1 ··· k and hence (8) shows that for any c> 0, b 1 1/c − c k log PN (a/b) = log max PN (a/b)+ O log A . 0 N 11 A simple reindexing shows that here b 1 b N 1 − − − 2 sin(πna/b) = 2 sin(π(b j)a/b) = Pb N 1(a/b). | | | − | − − j=1 n=YN+1 Y As observed in (6), we also have Pb 1(a/b)= b. Hence (21) yields − log PN (a/b) + log Pb N 1(a/b) = log b, − − as claimed. 2.2 The transfer principle Let α = [a0; a1, a2,... ] be an arbitrary irrational number with convergents pk/qk = [a0; a1,...,ak]. Let Ak = 1 + max1 ℓ k aℓ, and let x denote the distance from ≤ ≤ k k a real number x to the nearest integer. The sequence qk satisfies the recursion qk = akqk 1 + qk 2 with initial conditions q0 = 1, q1 = a1. Recall that for any − − k 1 and any 0 cot(πnα) (124 + 24 log Ak) qk. (23) ≤ n=1 X The same bound holds in the rational setting as well, i.e. N cot(πnpk/qk) (124 + 24 log Ak) qk. (24) ≤ n=1 X Indeed, we can apply (23) to a sequence of irrational α’s converging to pk/qk, whose continued fraction expansions have initial segments identical to pk/qk = [a0; a1,...,ak]. The same cotangent sum and various generalizations thereof in the rational setting have been studied recently in [7], and used in [6] to establish (2). Cotangent sums have a long history in analytic number theory; some of them are known to satisfy interesting reciprocity formulas, and they also appear in the Nyman–Beurling–B´aez-Duarte approach to the Riemann hypothesis. See [4, 5] for more details. 12 Proof of Proposition 3. We consider the cases qk < 200 and qk 200 sepa- rately, starting with the former; the value 200 is of course basically accidental.≥ First, assume that qk < 200. If k = 1 and a1 = 1, then N = 0 and we are done; we may therefore assume that either k 2, or k = 1 and a > 1. For any ≥ 1 0 sin(πnα) 50π 1 < , sin(πnp /q ) − a k k k+1 and hence 50π 200 P (α) 50π 200 1 N 1+ . − a ≤ P (p /q ) ≤ a k+1 N k k k+1 This finishes the proof in the case qk < 200. Next, assume that q 200. Observe that q q a for all 1 ℓ k; k ≥ k ≥ ℓ ≥ ℓ ≤ ≤ in particular, qk Ak 1. From the assumption qk 200 we thus deduce qk 200(A 1) ≥10√A−. Using a trigonometric identity,≥ we can write ≥ k − ≥ k p N N PN (α) sin(πnα) = = (1 + xn + yn) , (26) PN (pk/qk) sin(πnpk/qk) n=1 n=1 Y Y where x = cos(πn(α p /q )) 1, n − k k − y = sin(πn(α p /q )) cot(nπp /q ). n − k k k k Here pk 1 1 1 1 1 α < 2 1 qk 2 1 . − qk qkqk+1 ≤ q − +1 ≤ q − Ak +1 k qk−1 ! k 13 From the Taylor expansions of sine and cosine, for all 0 sin(πn(α p /q )) πn α p /q 1 1 y | − k k | | − k k| 1 | n| ≤ sin(π/q ) ≤ π/q π3/(6q3) ≤ 1 π2/(6q2) − A +1 k k − k − k k 1 1 3 1 1 . ≤ 1 π2/(600A ) − A +1 ≤ − 10A − k k k The previous two estimates give xn + yn 1 1/(4Ak); the point is that xn + yn is bounded away from 1. | | ≤ − Observe that for any− x 1 1/(4A ), | | ≤ − k x 2x2 log(4A ) x e − k 1+ x e . ≤ ≤ x+2x2 log(4A ) Indeed, one readily verifies that the function e− k (1+ x) attains its min- imum on the interval [ 1+1/(4A ), 1 1/(4A )] at x = 0. Applying this estimate − k − k with x = xn + yn in each factor of (26), we obtain N N PN (α) 2 2 = exp O (xn + yn) + (xn + yn) log(4Ak) . (27) PN (pk/qk) !! n=1 n=1 X X Note that the right-hand side of (27) provides both an upper and a lower bound for the quotient on the left-hand side. Since π2n2 p 2 π2 x α k , | n| ≤ 2 − q ≤ 2a2 q2 k k+1 k 2 the contribution of xn and xn in (27) is negligible: N N 1 log A x + x2 log(4A ) + k . | n| n k ≪ a2 q a4 q3 n=1 n=1 k+1 k k+1 k X X From Lubinsky’s bound on cotangent sums (24), summation by parts and 1 sin(π(n + 1)(α pk/qk)) sin(πn(α pk/qk)) π α pk/qk 2 , | − − − | ≤ | − |≪ ak+1qk we obtain N N log Ak yn = sin(πn(α pk/qk)) cot(πnpk/qk) . − ≪ ak+1 n=1 n=1 X X 14 Finally, N N q 1 log A k− log A log A y2 log(4A ) k k k , n k ≪ a2 q2 np /q 2 ≤ a2 q2 j/q 2 ≪ a2 n=1 n=1 k+1 k k k j=1 k+1 k k k+1 X X k k X k k since the integers npk, 1 n N attain each nonzero residue class modulo qk at most once. Hence (27) simplifies≤ ≤ as P (α) log A N = exp O k , P (p /q ) a N k k k+1 which proves the proposition. 3 Quadratic irrationals Fix a quadratic irrational α. Throughout this section, constants and implied constants depend only on α. The continued fraction expansion is of the form α = [a0; a1,...,as, as+1,...,as+p], where the overline means period. As before, pk/qk = [a ; a ,...,a ] denotes the k-th convergent to α; further, let δ =( 1)k(q α p ). 0 1 k k − k − k The sequences qk and pk satisfy the same recursion; consequently, δk = akδk 1 + − − δk 2 for all k 2. If k 1, or k = 0 and a1 > 1, then δk = qkα . − For any 1 ≥ r p, let≥ k k ≤ ≤ 0 1 0 1 Tr = . 1 as+r+p ··· 1 as+r+1 The recursion qk = akqk 1 + qk 2 can be written in the form − − m qs+r 1 qs+mp+r 1 T − = − , m =1, 2,... r q q s+r s+mp+r Observe that det T = ( 1)p, and that tr T does not depend on r. Therefore r − r the eigenvalues η and µ of Tr are the same for all 1 r p. Since qk exponentially fast, we have, say, η > 1 and µ = ( ≤1)p/η≤. Consequently,→ the ∞ − recursions for qk and δk have solutions m mp m qs+mp+r = Crη + Dr( 1) η− , m − (28) δs+mp+r = Erη− with some constants Cr, Er > 0 and Dr, 1 r p. In particular, log qk = λ(α)k + O(1) with λ(α) = log η1/p. ≤ ≤ 15 Lemma 1. For any k 0, we have κδk δk+1 (1 κ)δk with some constant κ> 0. ≥ ≤ ≤ − Proof. We claim that δk (ak+2 + 2)δk+1 for all k 0. Indeed, if k = 0, a1 =1 this can be verified “by hand”;≤ else, from (22) we obtain≥ 1 ak+2 +2 δk < < (ak+2 + 2)δk+1. qk+1 ≤ qk+2 + qk+1 On the other hand, 1 δk+1 ak+2δk+1 = δk δk+2 δk δk+1, ≤ − ≤ − ak+3 +2 and hence δ δ (a + 2)/(a + 3). The claim thus follows with κ = k+1 ≤ k k+3 k+3 1/(maxk 1 ak + 3). ≥ 3.1 Perturbed Sudler products The fundamental objects in the proof of Theorem 1 are “perturbed” versions of the Sudler product defined as qk P (α, x) := 2 sin(π(nα +( 1)kx/q )) , x R. qk | − k | ∈ n=1 Y Perturbed Sudler products were first introduced by Grepstad, Kaltenb¨ock and Neum¨uller [10], and have since been used in [1] and [13]. The relevance of these functions come from the Ostrowski expansion of integers, which we now recall. Any integer N 0 can be written in a unique way in the form N = ∞ b q , ≥ k=0 k k where 0 b0 a1 1 and 0 bk ak+1, k 1 are integers satisfying the extra ≤ ≤ − ≤ ≤ ≥ P rule that bk 1 = 0 whenever bk = ak+1. Of course, the series only has finitely many − nonzero terms; more precisely, if 0 N < q , then b = b = = 0. Given ≤ k+1 k+1 k+2 ··· an integer N 0 with Ostrowski expansion N = ∞ b q , let us introduce the ≥ k=0 k k notation ∞ P ε (N) := q ( 1)k+ℓb δ , k k − ℓ ℓ ℓ=Xk+1 ℓ where δℓ = ( 1) (qℓα pℓ) was already defined at the beginning of this section. Further, we shall− write−f(x)= 2 sin(πx) . | | Lemma 2. For any integer N 0 with Ostrowski expansion N = ∞ b q , ≥ k=0 k k bk 1 ∞ − P PN (α)= Pqk (α, bqkδk + εk(N)). kY=0 Yb=0 16 Proof. Note that only finitely many factors are different from 1. Let Nk = ∞ b q . Then N = N N N , and N = 0 for all large enough ℓ=k ℓ ℓ 0 ≥ 1 ≥ 2 ≥ ··· k k. By the definition of Sudler products, P Nk ∞ PN (α)= f(nα) n=N +1 kY=0 Yk+1 bkqk ∞ = f(nα + Nk+1α) n=1 kY=0 Y bk 1 qk ∞ − ∞ = f nα + bqkα + bℓqℓα n=1 ! kY=0 Yb=0 Y ℓ=Xk+1 bk 1 qk ∞ − ∞ = f nα +( 1)kbδ + ( 1)ℓb δ − k − ℓ ℓ n=1 ! kY=0 Yb=0 Y ℓ=Xk+1 bk 1 ∞ − = Pqk (α, bqkδk + εk(N)). kY=0 Yb=0 The main message of the next lemma is that for quadratic irrational α, the function Pqk (α, x) has a positive lower bound at all points which appear in the claim of Lemma 2. From now on let 1 [k] p denote the remainder of k s ≤ ≤ − modulo p, where s is the length of the pre-period in the continued fraction for α; that is, if k = s + mp + r, then [k]= r. Lemma 3. (i) For any integer N 0 with Ostrowski expansion N = k∞=0 bkqk, any k 0 and any 0 b < b≥ we have P (α, bq δ + ε (N)) 1 uniformly in N≥, k ≤ k qk k k k ≫ and b. P (ii) There exist compact intervals I , 1 r p, and a constant k >s with the r ≤ ≤ 0 following properties. First, for any integer N 0 with Ostrowski expansion ≥ N = ∞ b q , any k k and any 0 b < b we have bq δ +ε (N) I . k=0 k k ≥ 0 ≤ k k k k ∈ [k] Second, Pq (α, x) 1 on I[k] uniformly in k k0. P k ≫ ≥ Proof. Fix an integer N 0 with Ostrowski expansion ∞ b q , and integers ≥ k=0 k k k 0 and 0 b < bk. We necessarily have bk > 0; in particular, k 1, or k = 0 ≥ ≤ P ≥ 17 and a1 > 1. Observe that ∞ ε (N)= q ( 1)k+ℓb δ q (a δ + a δ + ) k k − ℓ ℓ ≤ k k+3 k+2 k+5 k+4 ··· ℓ=Xk+1 = q ((δ δ )+(δ δ )+ ) k k+1 − k+3 k+3 − k+5 ··· = qkδk+1. Since bk > 0, by the extra rule of the Ostrowski expansions we have bk+1 < ak+2. Therefore ∞ ε (N)= q ( 1)k+ℓb δ q ((a 1)δ + a δ + ) k k − ℓ ℓ ≥ − k k+2 − k+1 k+4 k+3 ··· ℓ=Xk+1 = q ( δ +(δ δ )+(δ δ )+ ) − k − k+1 k − k+2 k+2 − k+4 ··· = q (δ δ ). − k k − k+1 Letting κ> 0 be as in Lemma 1, we thus have εk(N) (1 κ)qkδk. Consider now | | ≤ − qk P (α, bq δ + ε (N)) = f((n + bq )α +( 1)kε (N)/q ). qk k k k k − k k n=1 Y For each 1 n q we have n + bq a q < q , and hence by the best ≤ ≤ k k ≤ k+1 k k+1 approximation property of continued fractions, (n + bq )α +( 1)kε (N)/q q α ε (N) /q κδ . k − k k ≥k k k−| k | k ≥ k Consequently, P (α, bq δ +ε (N)) 1 for any fixed k 0. It will thus be enough qk k k k ≫ ≥ to prove Lemma 3 (ii), and Lemma 3 (i) will follow. We now prove Lemma 3 (ii). Observe that (28) implies qs+mp+rδs+mp+r Br as m , where B = C E > 0, 1 r p, are constants. Define → →∞ r r r ≤ ≤ I = [ (1 κ/2)B , (a κ/2)B ] (1 r p). r − − r s+r+1 − r ≤ ≤ These intervals, together with some constant k0, to be chosen, satisfy the claim. Choosing k large enough, for all k k and all 0 b < a we have bq δ + 0 ≥ 0 ≤ k+1 k k [ (1 κ)q δ , (1 κ)q δ ] I . In particular, bq δ + ε (N) I for all N 0, − − k k − k k ⊆ [k] k k k ∈ [k] ≥ all k k0 and all 0 b < bk. Now≥ let k k ≤and x I be arbitrary, and let us prove a lower bound for ≥ 0 ∈ [k] Pqk (α, x). Then x = bB[k] + y for some appropriate integer 0 b < ak+1, and some y (1 κ/2)B . Let ≤ | | ≤ − [k] ( 1)k(y + b(B q δ )) z = − [k] − k k , qk 18 and note that (1 κ/2)B[k] +(ak+1 1) qkδk B[k] z − − | − | (1 κ/4)δk (29) | | ≤ qk ≤ − provided k0 was chosen large enough. With this choice of z we have f(nα +( 1)kx/q )= f((n + bq )α + z), − k k and so qk Pqk (α, x) n=1 f((n + bqk)α + z) = q (b+1)qk k f(nα) n=1 f((n + bqk)α) n=bqk+1 Q qk (30) Q Q = (cos(πz) + sin(πz) cot(π(n + bqk)α)) , n=1 Y where the last equation follows from standard trigonometric identit ies. Using (n + bq )α q α = δ and (29), we obtain k k k≥k k k k sin(πz) cos(πz) 1 + sin(πz) cot(π(n + bqk)α) cos(πz) 1 + | | | − |≤| − | sin(πδk) π2 π(1 κ/4)δ (1 κ/4)2δ2 + − k ≤ 2 − k πδ π3δ3/6 k − k 1 κ/8 ≤ − provided k0 was chosen large enough; the point is that each factor in (30) is bounded away from 0. Following the steps in the proof of Proposition 3 (in par- ticular, recalling the cotangent sum estimate (23)), we thus deduce that Pqk (α, x) (b+1)qk f(nα) n=bqk+1 qk qk Q 2 2 = exp O 1+ δk cot(π(n + bqk)α) + δk cot (π(n + bqk)α) !! n=1 n=1 X X = exp (O(1)) . On the other hand, a general result of Lubinsky [15, Proposition 5.1] implies that 1 PN (α) 1 whenever the Ostrowski expansion of N contains 1 nonzero terms.≪ In particular,≪ ≪ (b+1)q k P (α) f(nα)= (b+1)qk 1, Pbqk (α) ≫ n=Ybqk+1 and hence P (α, x) 1, as claimed. qk ≫ 19 3.2 The limit functions The perturbed Sudler products Pqk (α, x) were shown to converge to an explicitly given limit function for the special irrationals α = [0; a, a, a, . . . ] in [1]. A gen- eralization to all quadratic irrationals has recently been announced by Grepstad, Technau and Zafeiropoulos [13]; see also [12] for a version without the perturba- tion variable x. In this paper we prove the locally uniform convergence of Pqk (α, x) with an explicit rate. This explicit (in fact, exponential) rate is needed to derive the O(max 1, 1/c ) resp. O(1) error terms in Theorem 1. { } Let α = [a0; a1,...,as, as+1,...,as+p] be a quadratic irrational, and let Cr, Er be as in (28). For any 1 r p let ≤ ≤ 1 2 C E 2 ∞ r r nαr 2 x + 2 Gr(α, x)=2π x + CrEr 1 CrEr { } − , (31) | | − n − n2 n=1 Y where αr = [0; as+r+p,...,as+r+2, as+r+1]. Theorem 4. The infinite product in (31) is locally uniformly convergent on R. The function G (α, ) is continuous on R, and continuously differentiable on the r · open set x R : Gr(α, x) =0 . For any compact interval I R and any integer k 1, { ∈ 6 } ⊂ ≥ 1/2 3/4 2 P (α, x)= 1+ O q− log q G (α, x)+ O q− , x I qk k k [k] k ∈ with implied constants depending only on I and α. In particular, P (α, ) qs+mp+r · → G (α, ) locally uniformly on R, as m . r · →∞ We postpone the proof to Section 4. Note that the periodicity of the continued fraction expansion is crucial for such a limit relation; in particular, Theorem 4 does not hold for all badly approximable α. Lemma 3 implies that G (α, ) > 0, and consequently that log G (α, ) is r · r · Lipschitz on the compact interval Ir. The main idea of the proof of Theorem 1 is to replace the perturbed Sudler product Pqk (α, bqkδk + εk(N)) by its limit G (α, bq δ +ε (N)) in the claim of Lemma 2. To this end, for any integer N 0 [k] k k k ≥ with Ostrowski expansion N = k∞=0 bkqk let P bk 1 ∞ − GN (α)= G[k](α, bqkδk + εk(N)), kY=k0 Yb=0 where k0 is the constant from the conclusion of Lemma 3 (ii). 20 Lemma 4. For any real c> 0 and any integer ℓ 1, we have ≥ 1/c 1/c c c log PN (α) = log GN (α) + O(1), 0 N k0 1 ℓ 1 bk 1 P − − − PN (α) Pqk (α, bqkδk + εk(N)) = Pqk (α, bqkδk + εk(N)) . GN (α) · G[k](α, bqkδk + εk(N)) Yk=0 kY=k0 Yb=0 k0 1 Lemma 3 (i) shows that here 1 − P (α, bq δ + ε (N)) 1. Since by ≪ k=0 qk k k k ≪ Lemma 3 (ii) we have G[k](α, bqkδk + εk(N)) 1, each factor k0 k ℓ 1 also Q ≫ ≤ ≤ − stays between two positive constants. Finally, for a large enough constant k1 >k0 Theorem 4 gives ℓ 1 bk 1 ℓ 1 bk 1 − − − − Pqk (α, bqkδk + εk(N)) 1/2 3/4 = 1+ O qk− log qk G[k](α, bqkδk + εk(N)) kY=k1 Yb=0 kY=k1 Yb=0 ∞ 1/2 3/4 = 1+ O q− log q [1/2, 2]. k k ∈ kY=k1 Hence 1 P (α)/G (α) 1, and the claims follow. ≪ N N ≪ 3.3 Approximate additivity The final step is to prove that our sequences with PN (α) replaced by GN (α) are additive up to a small error; the proof of Theorem 1 will then be immediate. Lemma 5. For any real c> 0, the sequences 1/c c cm := log GN (α) and cm∗ := log max GN (α) 0 N 21 Proof. First, we prove that cm and cm∗ are approximately subadditive. Let 0 k0+(m+n)p 1 ≤ − N < qk0+(m+n)p be an integer with Ostrowski expansion N = k=0 bkqk. Consider the natural factorization P k0+mp 1 b 1 k0+(m+n)p 1 b 1 − k− − k− GN (α)= G[k](α, bqkδk + εk(N)) G[k](α, bqkδk + εk(N)). k=k0 b=0 k=k0+mp b=0 Y Y Y Y (32) k0+mp 1 k0+(m+n)p 1 − − Let us write N = N1 + N2 with N1 = k=0 bkqk and N2 = k=k0+mp bkqk. The plan of the proof is to show that, making a small error, we can replace εk(N) in (32) by ε (N ) and ε (N ), respectively,P so that G G GP . Then we will k 1 k 2 N ≈ N1 N2 show that we can replace N2 by a number N2∗ having the “shifted” Ostrowski k0+np 1 representation N = − b q , and obtain G G G ∗ . This approxi- 2∗ k=k0 k+mp k N N1 N2 mate shift-invariance is crucial for the argument, and comes≈ from the periodicity P of the continued fraction expansion of α. From GN GN GN ∗ we can deduce that ≈ 1 2 c c + c and c∗ c∗ + c∗ , which is what we want to prove. m+n ≈ m n m+n ≈ m n Now we make this precise. Note that for any k k k + mp 1, 0 ≤ ≤ 0 − k0+(m+n)p 1 k0+mp 1 − − ε (N) ε (N ) = q ( 1)k+ℓb δ q ( 1)k+ℓb δ | k − k 1 | k − ℓ ℓ − k − ℓ ℓ ℓ=k+1 ℓ=k+1 X X k0+(m+n)p 1 − q δ ≪ k ℓ ℓ=Xk0+mp q δ . ≪ k k0+mp Since log Gr is Lipschitz on Ir, the previous estimate implies that the first factor in (32) is k0+mp 1 b 1 − k− G[k](α, bqkδk + εk(N)) kY=k0 Yb=0 k0+mp 1 b 1 − k− O(qkδk +mp) = G[k](α, bqkδk + εk(N1))e 0 (33) kY=k0 Yb=0 k0+mp 1 b 1 − k− O(qk +mpδk +mp) = e 0 0 G[k](α, bqkδk + εk(N1)) kY=k0 Yb=0 O(1) = e GN1 (α). k0+np 1 Now let N = − b q ; observe that this is a valid Ostrowski expan- 2∗ k=k0 k+mp k sion of 0 N ∗ < qk +np. Using (28), for any k0 + mp k k0 +(m + n)p 1 and ≤ 2 P0 ≤ ≤ − 22 any 0 b < bk, ≤ 2m 2k/p bqkδk bqk mpδk mp η − . | − − − |≪ Similarly, εk(N) εk mp(N2∗) | − − | k0+(m+n)p 1 k0+np 1 − − k+ℓ k mp+ℓ = qk ( 1) bℓδℓ qk mp ( 1) − bℓ+mpδℓ − − − − ℓ=k+1 ℓ=k mp+1 X X− k0+(m+n)p 1 − k+ℓ = ( 1) bℓ (qkδℓ qk mpδℓ mp) − − − − ℓ=k+1 X k0+(m+n)p 1 − 2m k/p ℓ/p η − − ≪ ℓ=Xk+1 2m 2k/p η − . ≪ Therefore the second factor in (32) is k0+(m+n)p 1 b 1 − k− G[k](α, bqkδk + εk(N)) k=kY0+mp Yb=0 k0+(m+n)p 1 bk 1 − − − O(η2m 2k/p ) = G[k](α, bqk mpδk mp + εk mp(N2∗))e − − − k=kY0+mp Yb=0 k0+np 1 bk+mp 1 − − O(1) = e G[k](α, bqkδk + εk(N2∗)) kY=k0 Yb=0 O(1) ∗ = e GN2 (α). (34) From (32), (33) and (34) we finally obtain the approximate factorization O(1) GN (α) = GN (α)GN ∗ (α)e . As N runs in the interval 0 N < qk +(m+n)p, 1 2 ≤ 0 we obtain each pair (N1, N2∗) [0, qk0+mp) [0, qk0+np) at most once by the unique- ness of Ostrowski expansions.∈ Therefore × G (α)c G (α)c G (α)c eO(c), N ≤ N N 0 N 23 and the approximate subadditivity cm+n cm+cn+O(1) and cm∗ +n cm∗ +cn∗ +O(1) follows. ≤ ≤ The proof of approximate superadditivity is entirely analogous. Let 0 ≤ N ′ < qk0+mp and 0 N ′′ < qk0+np be integers with Ostrowski expansions k0+mp 1 ≤ k0+np 1 k0+(m+n+1)p 1 N ′ = k=0 − bk′ qk and N ′′ = k=0 − bk′′qk. Define N = k=0 − bkqk, where P P P b′ if 0 k k + mp 1, k ≤ ≤ 0 − b = 0 if k + mp k k +(m + 1)p 1, k 0 ≤ ≤ 0 − bk′′ (m+1)p if k0 +(m + 1)p k k0 +(m + n + 1)p 1. − ≤ ≤ − Note that this is a valid Ostrowski expansion of an integer 0 N < q ; ≤ k0+(m+n+1)p the block of zeroes in the middle ensures that the extra rule (bk 1 = 0 whenever − bk = ak+1) is satisfied. Repeating the arguments from above, we deduce the approximate factorization ′ ′′ O(1) GN (α)= GN (α)GN (α)e . Observe that as (N ′, N ′′) [0, qk0+mp) [0, qk0+np), we obtain each integer N [0, q ) at most O∈(1) times; indeed,× from ∈ k0+(m+n+1)p the value of N one can recover all Ostrowski digits of N ′ and N ′′ except for bk′′, 0 k k 1. Therefore ≤ ≤ 0 − G (α)c G (α)c G (α)ceO(c), N N ≤ N 0 N Proof of Theorem 1. According to Lemma 5, the sequence cm+L is subadditive, and the sequence c L is superadditive with some constant L = O(max 1, 1/c ). m − { } Fekete’s subadditive lemma thus shows that cm/m converges, and its limit is cm cm + L cm L pKc(α) := lim = inf = sup − . m m m 1 m m 1 m →∞ ≥ ≥ 24 The previous relations also yield the rate of convergence cm = Kc(α)mp + O(max 1, 1/c ). Since K (α)= O(max 1, 1/c ) (see (9)), we have { } c { } 1/c log G (α)c = K (α)k + O(max 1, 1/c ). N c { } 0 N 1/c log P (p /q )c = K (α)k + O(max 1, 1/c ), N k k c { } 0 N log max PN (pk/qk)= K (α)k + O(1). 0 N It follows from (7) and (9) that the constants Kc(α) and K (α) are positive. ∞ 4 Locally uniform convergence of perturbed Sudler products Proof of Theorem 4. Fix a compact interval I R. Throughout the proof, con- stants and implied constants depend only on I and⊆ α. For the sake of readability, we continue to write f(x)= 2 sin(πx) . | | We start by peeling off the last factor in Pqk (α, x) and using α = pk/qk + ( 1)kδ /q to get − k k q 1 k− np nδ + x P (α, x)= f(q α +( 1)kx/q ) f k +( 1)k k qk k − k q − q n=1 k k q 1 Y k− np n 1 2x + q δ = f(δ + x/q ) f k +( 1)k δ +( 1)k k k . k k q − q − 2 k − 2q n=1 k k k Y The factors in the previous line depend only on the remainder of n modulo qk. The k general identity qkpk 1 pkqk 1 =( 1) in the theory of continued fractions shows − − − − k+1 that qk and qk 1 are relatively prime, and pkqk 1 ( 1) (mod qk). Reordering − − ≡ − the product via the bijection n qk 1n of the set of nonzero residues modulo qk, − and using the identity (6), we obtain7→ nqk−1 2x+q δ qk 1 n 1 k k − f q q 2 δk 2q P (α, x)= f(δ + x/q )q k − k − − k . qk k k k n f(on/q ) n=1 k Y 25 Let us combine the nth and the (qk n)th factors in the product via the trigono- −2 2 metric identity f(u v)f(u + v)= f (u) f (v) . If qk is even, then the n = qk/2 2 − | − | factor is 1 + O(qk− ), thus 2 Pqk (α, x) =(1+ O(qk− ))f(δk + x/qk)qk 2 n nqk−1 1 2 2x+qkδk f q q 2 δk f 2q (35) k − k − − k . × n o 2 n 0 2 n nqk−1 1 2 2x+qkδk f δk f qk − qk − 2 − 2qk n o n f 2 qk 2 n nqk−1 1 f δk qk − qk − 2 1 = + O 2 nf 2 n o n qk nqk 1 1 n 1 =1 sin 2π − δ cot π + O . − q − 2 k q n2 k k In particular, each factor ψ(q ) n < q /2 is 1 + O(1/n), and thus k ≤ k 2 n nqk−1 1 2 2x+qkδk f δk f qk − qk − 2 − 2qk n o 2 n ψ(q ) n 26 Observe also that cot(πn/qk) is monotone in 0 nqk 1 1 n log qk sin 2π − δk cot π , qk − 2 qk ≪ ψ(qk) ψ(qk) n 2 n nqk−1 1 2 2x+qkδk f δk f qk qk 2 2qk log q − − − =1+ O k . n o 2 n ψ(qk) ψ(qk) n 27 so replacing qk 1/qk by α[k] introduces a negligible multiplicative error of 1 + 2 − O(qk− ψ(qk)). We have thus deduced 2 n nqk−1 1 2 2x+qkδk f δk f qk − qk − 2 − 2qk n o 2 n N Lemma 6. For any integers N < N N and any 1 r p, 0 1 ≤ 2 ≤ ≤ 2 2 N2 nα 1 x + CrEr log N { r} − 2 2 1 1 CrEr 2 =1+ O . − n − n N1 n=N1 Y Proof. Choosing the constant N0 large enough, each factor stays in, say, [1/2, 2]. Since the partial quotients of αr are bounded by a constant depend- ing only on α, the same discrepancy estimate and Koksma’s inequality yield N ( nα 1/2) log N for all N 1. Applying summation by parts, n=1 { r} − ≪ ≥ we deduce P N2 1 nαr log N1 { } − 2 , n ≪ N1 n=N1 X and the claim of Lemma 6 follows. Lemma 6 immediately implies via the Cauchy criterion that the infinite product 1 2 C E 2 ∞ r r nαr 2 x + 2 1 CrEr { } − (38) − n − n2 n=N0+1 Y is uniformly convergent on I, and its limit is positive. Its logarithm is given by a uniformly convergent series; the series of term-by-term derivatives 2 1 1 2 CrEr − ∞ nαr x + 2x C E 1 C E { } − 2 2 − − r r − r r n − n2 n2 n=N0+1 ! X is also seen to be uniformly convergent on I. Therefore the logarithm of the infinite product in (38) is continuously differentiable on I; clearly so is the infinite product itself. Multiplying by the missing factors 1 n N , it follows that the infinite ≤ ≤ 0 28 product (31) defining Gr(α, x) is uniformly convergent on I, Gr(α, x) is continuous on I, and continuously differentiable on I except at its (finitely many) zeroes. This proves all claims on Gr(α, x). Repeating the arguments above for the factors 1 n N , we deduce ≤ ≤ 0 N 2 n nqk−1 1 2 2x+qkδk 0 f δk f qk − qk − 2 − 2qk f(δk + x/qk)qk n o 2 n n=1 f Y qk 2 C[k]E[k] N0 1 2 x + nα[k] 2 2 2 =2π x + C E 1 C E { } − + O(q− ) [k] [k] [k] [k] 2 k | | − n − n n=1 Y with an additive instead of multiplicative error term, since the factor s are not bounded away from zero. Lemma 6 also shows that the product on the right hand side of (37) can be extended to all N0 < n up to a negligible error, therefore combining (35)–(37) and the previous formula, we obtain 3 log qk ψ(qk) 2 P (α, x)= 1+ O + G (α, x)+ O(q− ). qk ψ(q ) q2 [k] k k k The optimal choice is ψ(t)= t1/2 log1/4 t. This finishes the proof of Theorem 4. Acknowledgements CA is supported by the Austrian Science Fund (FWF), projects F-5512, I-3466, I-4945 and Y-901. BB is supported by FWF project Y-901. We want to thank Agamemnon Zafeiropoulos for drawing our attention to the papers of Bettin and Drappeau, and for keeping us informed about his joint work with Grepstad and Neum¨uller. References [1] C. Aistleitner, N. Technau and A. Zafeiropoulos: On the order of magnitude of Sudler products. arXiv:2002.06602. [2] J. Andersen and S. Hansen: Asymptotics of the quantum invariants for surgeries on the figure 8 knot. J. Knot Theory Ramifications 15 (2006), 479–548. [3] A. Avila and S. Jitomirskaya: The Ten Martini Problem. Ann. of Math. 170 (2009), 303– 342. [4] S. Bettin and B. Conrey: Period functions and cotangent sums. Algebra Number Theory 7 (2013), 215–242. 29 [5] S. Bettin, and J. B. Conrey: A reciprocity formula for a cotangent sum. Int. Math. Res. Not. IMRN 24 (2013), 5709–5726. [6] S. Bettin and S. Drappeau: Modularity and value distribution of quantum invariants of hyperbolic knots. arXiv:1905.02045. [7] S. Bettin and S. Drappeau: Partial sums of the cotangent function. J. Th´eor. Nombres Bordeaux 32 (2020), 217–230. n ak [8] P. Erd˝os and G. Szekeres: On the product k=1(1 z ). Acad. Serbe Sci. Publ. Inst. Math. 13 (1959), 29–34. − Q [9] S. Garoufalidis and D. Zagier: Quantum modularity of the Kashaev invariant. In prepara- tion. [10] S. Grepstad, L. Kaltenb¨ock and M. Neum¨uller: A positive lower bound for N lim infN→∞ 2 sin πrφ . Proc. Amer. Math. Soc. 147 (2019), 4863–4876. r=1 | | [11] S. Grepstad,Q L. Kaltenb¨ock and M. Neum¨uller: On the asymptotic behaviour of the sine n product r=1 2 sin πrα . In: Discrepancy Theory. Radon Series on Computational and Applied Mathematics,| 26.| De Gruyter, 2020. Q [12] S. Grepstad and M. Neum¨uller: Asymptotic behaviour of the Sudler product of sines for quadratic irrationals. J. Math. Anal. Appl. 465 (2018), 928–960. [13] S. Grepstad, M. Neum¨uller and A. Zafeiropoulos: On the order of magnitude of Sudler products II. In preparation. [14] L. Kuipers and H. Niederreiter: Uniform Distribution of Sequences. Pure and Applied Mathematics. Wiley-Interscience, New York-London-Sydney, 1974. [15] D. Lubinsky: The size of (q; q)n for q on the unit circle. J. Number Theory 76 (1999), 217–247. [16] H. Murakami and Y. Yokota: Volume conjecture for knots. Springer Briefs in Mathematical Physics, 30. Springer, Singapore, 2018. [17] C. Sudler Jr.: An estimate for a restricted partition function. Quart. J. Math. Oxford Ser. 15 (1964), 1–10. [18] D. Zagier: Quantum modular forms. Quanta of maths, Clay Math. Proc. pp. 659–675. Amer. Math. Soc. Providence, RI, 2010. 30