Welcome to the new version of CaltechAUTHORS. Login is currently restricted to library staff. If you notice any issues, please email coda@library.caltech.edu
Published January 10, 2017 | Submitted
Journal Article Open

Refinements of Gál's theorem and applications


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].

Additional Information

© 2016 Elsevier Inc. Received 25 February 2016, Revised 16 August 2016, Accepted 7 September 2016, Available online 30 September 2016. The first author is supported by a National Science Foundation postdoctoral fellowship, DMS-12042 and the Institute for Advanced Study Fund for Mathematics, the second author was partially supported by National Science Foundation grant DMS-1001068. The first author would like to thank Jeff Vaaler for first bringing Gál's theorem to his attention and many discussions on it and related topics. We would like to thank the referees for a very careful reading of the paper that improved its quality.

Attached Files

Submitted - 1408.2334.pdf


Files (221.5 kB)
Name Size Download all
221.5 kB Preview Download

Additional details

August 22, 2023
October 18, 2023