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Littlewood and Duffin--Schaeffer-Type Problems in Diophantine

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Littlewood and Duffin--Schaeffer-Type Problems in Diophantine LITTLEWOOD AND DUFFIN{SCHAEFFER-TYPE PROBLEMS IN DIOPHANTINE APPROXIMATION SAM CHOW AND NICLAS TECHNAU Dedicated to Andy Pollington Abstract. Gallagher's theorem describes the multiplicative diophantine approximation rate of a typical vector. We establish a fully-inhomogeneous version of Gallagher's theorem, a diophantine fibre refinement, and a sharp and unexpected threshold for Liouville fibres. Along the way, we prove an inhomogeneous version of the Duffin–Schaeffer conjecture for a class of non-monotonic approximation functions. Table of Contents 1. Introduction 2 1.1. Main results 6 1.2. Key ideas and further results 9 1.3. Open problems 14 1.4. Organisation and notation 17 2. Preliminaries 18 2.1. Continued fractions 18 arXiv:2010.09069v1 [math.NT] 18 Oct 2020 2.2. Ostrowski expansions 20 2.3. Bohr sets 23 2.4. Measure theory 25 2.5. Real analysis 25 2.6. Geometry of numbers 27 2020 Mathematics Subject Classification. 11J83, 11J54, 11H06, 52C07, 11J70. Key words and phrases. Metric diophantine approximation, geometry of numbers, additive combinatorics, continued fractions. 1 2 SAM CHOW AND NICLAS TECHNAU 2.7. Primes and sieves 30 3. A fully-inhomogeneous version of Gallagher's theorem 32 3.1. Notation and reduction steps 32 3.2. Divergence of the series 34 3.3. Overlap estimates, localised Bohr sets, and the small-GCD regime 40 3.4. Large GCDs 47 3.5. A convergence statement 52 4. Liouville fibres 54 4.1. A special case 54 4.2. Diophantine second shift 56 4.3. Liouville second shift 61 4.4. Rational second shift 65 5. Obstructions on Liouville fibres 65 Appendix A. Pathology 70 References 72 1. Introduction This manuscript concerns two fundamental problems in diophantine approxi- mation. We introduce a method to tackle, in a general context, inhomogeneous versions of Littlewood's conjecture which are metric in at least one parameter. Further, we prove an inhomogeneous version of the Duffin–Schaeffer conjecture for a relatively large class of functions. Let us begin by explaining the link to Littlewood's conjecture, and defer elaborating on the inhomogeneous Duffin–Schaeffer conjecture to Section 1.2. Around 1930, Littlewood raised the question of whether planar badly approx- 2 imable vectors exist in a multiplicative sense: That is, if for all (α1; α2) 2 R , we have lim inf nknα1k · knα2k = 0; (1.1) n!1 where k·k denotes the distance to the nearest integer. Until the time of writing, finding non-trivial examples of (α1; α2) satisfying (1.1), barring rather special LITTLEWOOD AND DUFFIN{SCHAEFFER-TYPE PROBLEMS 3 cases, evades the best efforts of the mathematicalp p community. For instance, the problem remains open even for (α1; α2) = ( 2; 3). Here non-trivial means that α1; α2 lie in the set Bad = fβ 2 R : 9c>0 nknβk > c for all n 2 Ng of badly approximable numbers. This set has Lebesgue measure zero, by the Borel{Cantelli lemma, but full Hausdorff dimension, by the Jarn´ık–Besicovitch theorem [12, Theorem 3.2]. Remark 1.1. By Dirichlet's approximation theorem, the inequality nknβk < 1 holds infinitely often for any β 2 R. So badly approximable numbers are pre- cisely those numbers for which Dirichlet's approximation theorem is optimal, up to a constant. The study of the measure theory and fractal geometry centring around (1.1) has turned out to be a fruitful endeavour. Indeed, it has led to various exciting developments in homogeneous dynamics and in diophantine approximation. We presently expound upon this. Homogeneous dynamics. There is a classical correspondence|the Dani corre- spondence|between the diophantine properties of a given vector in Rk, and the dynamical properties of the associated orbit of a lattice in the space SLk+1(R)=SLk+1(Z); under the group of diagonal matrices in R(k+1)×(k+1) having determinant 1. By virtue of this, Littlewood's conjecture is closely linked to a conjecture of Mar- gulis [43, Conjecture 1]. If true, Margulis' conjecture would imply Littlewood's conjecture. With this dynamical perspective, Einsiedler, Katok, and Lindenstrauss [24] established, inter alia, the striking result that the set of putative counterexam- ples to Littlewood's conjecture (1.1) has Hausdorff dimension zero. A crucial ingredient was a deep result of Ratner. From a similar departure point, Shapira [50] established a measure-theoretic, uniform version of an inhomogeneous Littlewood-type problem, solving an old problem of Cassels. To state it, we stress that `for almost all' (and similarly for `almost every') is in this manuscript always means with respect to the Lebesgue measure on the ambient space, expressing that the complement of the set under consideration has Lebesgue measure zero. Shapira proved that for almost all (α; β) 2 R2 the relation lim inf nknα − γk · knβ − δk = 0 n!1 holds for any γ; δ 2 R. Gorodnik and Vishe [31] improved this by a factor of (log log log log log n)λ, for some constant λ > 0. 4 SAM CHOW AND NICLAS TECHNAU While the above results suggest that Littlewood's conjecture might be correct, there is an indication against it: Adiceam, Nesharim, and Lunnon [1] proved that a certain function field analogue of the Littlewood conjecture is false. In what follows, the word metric is used to mean `measure-theoretic', as is cus- tomary in this area. The goal of metric number theory is to classify behaviour up to exceptional sets of measure zero. Metric multiplicative diophantine approximation. Alongside the theory of ho- mogeneous dynamics linked to Littlewood's conjecture, there have been signif- icant advances towards the corresponding metric theory. The first systematic result in this direction is a famous theorem of Gallagher [29]:1 For any non- k increasing : N ! R>0 and almost all (α1; : : : ; αk) 2 R , we have knα1k · · · knαkk < (n) (1.2) infinitely often if the relevant series of measures diverges, that is, if X k−1 (n)(log n) = 1: (1.3) n>1 2 Consequently (1.1) holds true for almost every (α1; α2) 2 R by a log-squared margin: 2 lim inf n(log n) knα1k · knα2k = 0: n!1 Continuing this line of research, Pollington and Velani [47] showed the following fibre statement, by exploiting the Fourier decay property of a certain fractal measure: If α1 2 Bad then there is a set of numbers α2 2 Bad, of full Hausdorff dimension, such that n log nknα1k · knα2k < 1 holds for infinitely many n 2 N. A further fibre statement concerning (1.1) was established by Beresnevich, Haynes, and Velani in [11, Theorem 2.4]. To state it, we recall that Liouville numbers are irrational real numbers α such that for any w > 0 the inequality knαk < n−w holds infinitely often. We denote the set of Liouville numbers by L. 1In fact Gallagher stated the theorem in a slightly weaker form, namely that k lim inf n(log n) knα1k · · · knαkk = 0 n!1 for almost all (α1; : : : ; αk), but cognoscenti will recall that the proof carries through com- fortably to give the divergence part of the version above, and that the convergence part was already known, see [14, Remark 1.2]. Gallagher's theorem is commonly referred to in one of three forms: the weakened divergence part, the full-strength divergence part, and the complete Lebesgue theory. We will refer to all three versions as Gallagher's theorem. LITTLEWOOD AND DUFFIN{SCHAEFFER-TYPE PROBLEMS 5 Theorem 1.2 ([11, Theorem 2.4]). Let α1 2 R n (Q [L) and γ 2 R. If the Duffin–Schaeffer conjecture is true, then for almost all α2 2 R we have 2 lim inf n(log n) knα1 − γk · knα2k = 0: n!1 When this theorem was proved, the Duffin–Schaeffer conjecture was still open. The former was then proved, without appealing to the Duffin–Schaeffer con- jecture, by the first named author [18] in a stronger form: Theorem 1.3 ([18]). Let α1; γ 2 R and assume that α1 2 R n (Q [L). If : N ! R>0 is non-increasing and the series X (n) log n (1.4) n>1 diverges, then for almost all α2 2 R there exist infinitely many n 2 N such that knα1 − γk · knα2k < (n): The proof used the structural theory of Bohr sets (see Section 1.2), as well as continued fractions and the geometry of numbers, in a crucial way. This combinatorial{geometric method was further developed by the authors [20] to prove higher-dimensional results, removing the reliance on continued fractions. Instead, a more versatile framework from the geometry of numbers was brought to bear on the problem. Another interesting facet of the approach is the application of diophantine transference inequalities [13, 16, 17, 19, 30, 38] to deal with the inhomogeneous shifts. By fixing α1 above, one considers pairs (α1; α2) which lie on a vertical line in the plane. With Yang, the first named author showed in [21] that if L0 is an arbitrary line in the plane then for almost all (α1; α2) 2 L0 we have 2 lim inf n(log n) knα1k · knα2k = 0: n!1 Subject to a generic diophantine assumption on L0, it was also shown there that if : N ! [0; 1) is non-increasing and X (n) log n < 1 n>1 then for almost all (α1; α2) 2 L0 the inequality knα1k · knα2k < (n) has at most finitely many solutions n 2 N. The divergence statement was attained via an effective asymptotic equidistribution theorem for unipotent orbits in SL3(R)=SL3(Z), whilst the convergence statement involved the corre- spondence between Bohr sets and generalised arithmetic progressions.
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