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Narrow progressions in the primes Terence Tao and Tamar Ziegler Abstract In a previous paper of the authors, we showed that for any polynomials P1,...,Pk Z[m] with P1(0)= = Pk(0) and any subset A of the primes in [N]= 1,...,N ∈of relative density at least··· δ > 0, one can find a “polynomialprogression” { } o(1) a+P1(r),...,a+Pk(r) in A with 0 < r N , if N is sufficiently large depending | |≤ on k,P1,...,Pk and δ. In this paper we shorten the size of this progression to 0 < L r log N, where L depends on k,P1,...,Pk and δ. In the linear case Pi = (i 1| )|≤m, we can take L independent of δ. The main new ingredient is the use of the− densification method of Conlon, Fox, and Zhao to avoid having to directly correlate the enveloping sieve with dual functions of unbounded functions. 1 Introduction 1.1 Previous results We begin by recalling the well-known theorem of Szemer´edi [18] on arithmetic progressions, which we phrase as follows: Theorem 1 (Szemeredi’s´ theorem). Let k 1 and δ > 0, and suppose that N is sufficiently large depending on k,δ. Thenanysubset≥ A of [N] := n Z :1 n N with cardinality A δN will contain at least one arithmetic{ progression∈ ≤ a≤,a +} | |≥ arXiv:1409.1327v2 [math.NT] 10 Oct 2014 r,...,a + (k 1)r of length k, with r > 0. − Terence Tao UCLA Department of Mathematics, 405 Hilgard Ave, Los Angeles CA 90095, USA, e-mail: [email protected] Tamar Ziegler Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel e-mail: [email protected] 1 2 Terence Tao and Tamar Ziegler In fact, by partitioning [N] into intervals of some sufficiently large but constant size L = L(k,δ) and using the pigeonhole principle, one can ensure that the progres- sion described above is “narrow” in the sense that r L(k,δ). In [10], Szemer´edi’s theorem was relativized to the≤ primes P = 2,3,5,7,... : { } Theorem 2 (Szemeredi’s´ theorem in the primes). Let k 2 and δ > 0, and suppose that N is sufficiently large depending on k,δ. Then≥ any subset A of [N] P with A δ [N] P will contain at least one arithmetic progression a,a∩+ r,...,a +| (|k ≥ 1)|r of length∩ | k, with r > 0. − In particular, the primes contain arbitrarily long arithmetic progressions. The proof of Theorem 2 does not place a bound on r beyond the trivial bound r N. In contrast with Theorem 1, one cannot hope here to make the step size r of the≤ progression as short as L(k,δ). Indeed, as observed in [1], if N is large enough P 1 P then from [8, Theorem 3] one can find a subset A of [N] with A 2 [N] (say) such that the gap between any two consecutive elements∩ of A| is|≥ clog| N∩for| some absolute constant c > 0, which of course implies in this case that≥ r must be k 1 at least clogN as well. Indeed, one can improve this lower bound to log − N by a small modification of the argument: Proposition 1. Let k 2, let ε > 0 be sufficiently small depending on k, and suppose that N is sufficiently≥ large depending on k. Then there exists a subset A of [N] P with A [N] P such that A does not contain any arithmetic progression a,∩a + | | ≫ | ∩ | k 1 r,...,a + (k 1)r of length k with 0 < r < ε log − N. − k 1 Proof. For any 0 < r < ε log − N, let Nr denote the number of a [N] such that a,a + r,...,a + (k 1)r are all prime. By the union bound and the∈ prime number theorem, it will suffice− to show that 1 N ∑ Nr k 1 ≤ 2 logN 0<r<ε log − N (say), since one can then form A by removing all the elements a in [N] P that are ∩ associated to one of the Nr. But from standard sieve theoretic bounds (see e.g. [12, Theorem 5.7]) we have N Nr CkG(k,r) ≤ logk N 1 where Ck depends only on k and G(k,r) is the singular series k 1 νp(r) G(k,r)= ∏ 1 1 (1) − p − p p and νp(r) is the number of residue classes a Z/pZ such that at least one of a,a + r,...,a + (k 1)r is equal to 0 mod p, so it suffices∈ to show that − 1 All sums and products over p in this paper are understood to be ranging over primes. Narrow progressions in the primes 3 ∑ G(k,r) CkM 0<r<M ≤ for any M 1 (we allow Ck to represent a different k-dependent constant from line to line, or even≥ within the same line). One could obtain precise asymptotics on the left-hand side using the calculations of Gallagher [7], but we can obtain a crude upper bound that suffices as follows. From (1) we see that 1 G(k,r) Ck exp Ck ∑ ≤ p r p! | and hence by [19, Lemma E.1] logCk p G(k,r) Ck ∑ ≤ p r p | and thus logCk p M ∑ G(k,r) Ck ∑ CkM 0<r<M ≤ p p p ≤ as required. In the conversedirection, if we use the “Cram´er randommodel” of approximating the primes P [N] by a random subset of [N] of density 1/logN, we can asymptot- ically almost∩ surely match this lower bound, thanks to the work of Conlon-Gowers [6] and Schacht [17]: Proposition 2. Let k 2 and δ,ε > 0, let C > 0 be sufficiently large depending on δ,k, and suppose that≥ N is sufficiently large depending on k,δ,ε. Let P [N] be chosen randomly, such that each n [N] lies in P with an independent probability⊂ of 1/logN. Then with probability at∈ least 1 ε, every subset A of P with A δ P will contain an arithmetic progression a,a +−r,...,a + (k 1)r of length k| with|≥ 0|<| k 1 − r C log − N. ≤ We remark that a modification of the argument in Proposition 1 shows that we cannot replace the large constant C here by an arbitrarily small constant c > 0. C k 1 Proof. We partition [N] into intervals I1,...,Im of length between 2 log − N and k 1 2N C log − N, thus m k 1 . For each interval Ii, we see from [17, Theorem 2.2] C log − N ≤ δε or [6, Theorem 1.12] that with probability at least 1 10 , every subset Ai of P Ii δ − ∩ with Ai P Ii will contain an arithmetic progression of length at least k. Call | |≥ 2 | ∩ | an interval Ii bad if this property does not hold, thus each Ii is bad with probability at most δε/10. By linearity of expectation followed by Markov’s inequality, we ε δ N conclude that with probability at least 1 , at most k 1 of the Ii are bad. Then − 5C log − N if A P is such that A δ P , then at most δ P of the elements of A are contained ⊂ | |≥ | | 2 | | in bad intervals, so from the pigeonhole principle there exists a good interval Ii such δ that A Ii P Ii , and the claim follows. | ∩ |≥ 2 | ∩ | 4 Terence Tao and Tamar Ziegler It is thus natural to conjecture that in Theorem 2 one can take r to be as small as k 1+o(1) log − N. Remark 1. If one seeks progressions inside the full set P of primes, rather than of dense subsets of the primes, then the Hardy-Littlewood prime tuples conjecture [13] predicts that one can take r to be of size Ok(1); indeed, one should be able to take r to be the product of all the primes less than or equal to k. In the case k = 2, the claim that one can take r = O(1) amounts to finding infinitely many bounded gaps between primes, a claim that was only recently established by Zhang [20]. For higher k, the claim r = Ok(1) appears to currently be out of reach of knownmethods; the best known result in this direction, due to Maynard [15] (and also independently in unpublished work of the first author), shows that for any sufficiently large R > 1, there exist infinitely many intervals of natural numbers of length R that contain clogR primes for some absolute constant c > 0, but this is too sparse a set of primes≥ to expect to find length k progressions for any k 3. ≥ We now consider generalizations of the above results, in which arithmetic pro- gressions a,a + r,...,a + (k 1)r are replaced by “polynomial progressions” a + − P1(r),...,a + Pk(r). More precisely, let Z[m] denote the ring of polynomials of one indeterminate variable m with integer coefficients. Then Bergelson and Leibman [3] established the following polynomial version of Theorem 1: Theorem 3 (Polynomial Szemeredi’s´ theorem). Let k 1,letP1,...,Pk Z[m] be ≥ ∈ such that P1(0)= = Pk(0), let δ > 0, and suppose that N is sufficiently large ··· depending on k,P1,...,Pk,δ. Then any subset A of [N] with cardinality A δN | |≥ will contain at least one polynomial progression a + P1(r),a + P2(r),...,a + Pk(r) with r > 0. Of course, Theorem 1 is the special case of Theorem 3 when Pi = (i 1)m.
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