Refinements of Gál's theorem and applications

Abstract

Abstract. We give a simple proof of a well-known theorem of Gál and of the recent related results of Aistleitner, Berkes and Seip [1] regarding the size of GCD sums. In fact, our method obtains the asymptotically sharp constant in Gál's theorem, which is new. Our approach also gives a transparent explanation of the relationship between the maximal size of the Riemann zeta function on vertical lines and bounds on GCD sums; a point which was previously unclear. Furthermore we obtain sharp bounds on the spectral norm of GCD matrices which settles a question raised in [2]. We use bounds for the spectral norm to show that series formed out of dilates of periodic functions of bounded variation converge almost everywhere if the coefficients of the series are in L 2 (log log 1/L) γ , with γ > 2. This was previously known with γ > 4, and is known to fail for γ < 2. We also develop a sharp Carleson-Hunt-type theorem for functions of bounded variations which settles another question raised in [1]. Finally we obtain almost sure bounds for partial sums of dilates of periodic functions of bounded variations improving [1]. This implies almost sure bounds for the discrepancy of {n k x} with n k an arbitrary growing sequences of integers.

Additive energy and the Hausdorff dimension of the exceptional set in metric pair correlation problems

“…Replacing these indicator functions by their respective Fourier series and using a combinatorial argument together with the orthogonality of the trigonometric system, we will reduce the problem of estimating the moments of such sums of dilated functions to a problem involving a certain GCD (greatest common divisors) sum. The role of GCD sums in metric number theory goes back at least to Koksma [19] (see also [20]), and they play a role in the context of the Duffin-Schaeffer conjecture in metric Diophantine approximation (see Dyer and Harman [13]) and in the theory of almost everywhere convergence of sums of dilated functions [2,22]. We will need the following upper bound of Bondarenko and Seip [8] for such GCD sums.…”

“…By the growth condition on a(x) and by (10) we have (22) g ′ m ∞ ≤ K2 m N d+5 for some universal positive constant K. Note also that…”

Sommes de Gál et applications

“…Un calcul standard, effectué dans lorsque , fournit Il suit, pour , …”

“…Gál a résolu la conjecture d'Erdős correspondante dans le cas . De nombreux articles récents [, , ] concernent le comportement asymptotique de la quantité elles‐même liée aux majorations de certains polynômes de Dirichlet et à celles de maximums localisés de la fonction zêta de Riemann sur la droite verticale d'abscisse .…”

Note on the Resonance Method for the Riemann Zeta Function

“…We refer to (4.1) and (4.2) as Gál-type sums because the topic begins with a sharp bound of Gál [11] (of order CN (log log N ) 2 ) for the growth of (4.1) when σ = 1. Dyer and Harman [9] obtained the first nontrivial estimates for the range 1/2 ≤ σ < 1, and during the past few years, we have reached an essentially complete understanding for the full range 0 < σ ≤ 1, thanks to the papers [2,4,5,16]. The techniques used for different values of σ differ considerably, and the problem is particularly delicate for σ = 1/2 at which an interesting "phase transition" occurs.…”