Refinements of G\'al's theorem and applications

TL;DR

This paper provides a simplified proof of Gál's theorem with the sharp constant, clarifies the link between the Riemann zeta function and GCD sums, and improves bounds on series convergence and discrepancy for functions of bounded variation.

Contribution

It introduces a new, transparent proof of Gál's theorem with the sharp constant and establishes sharp bounds on GCD matrices and series convergence for functions of bounded variation.

Findings

Obtained asymptotically sharp constant in Gál's theorem.

Established sharp bounds on the spectral norm of GCD matrices.

Proved almost everywhere convergence of series of dilates of bounded variation functions with improved conditions.

Abstract

We give a simple proof of a well-known theorem of G\'al 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 G\'al'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 $L^2 (\log\log 1/L)^{\gamma}$, with $\gamma > 2$. This was previously known with $\gamma >4$, and is known to fail for $\gamma<2$. 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.