Refinements of Gal's theorem and applications

We give a simple proof of a well-known theorem of Gal 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 Gal'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]. (C) 2016 Elsevier Inc. All rights reserved.