Refinements of Gál's theorem and applications

Mark Lewko*, Maksym Radziwiłł

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

15 Scopus citations


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 L2(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].

Original languageEnglish (US)
Pages (from-to)280-297
Number of pages18
JournalAdvances in Mathematics
StatePublished - Jan 10 2017


  • Carleson–Hunt theorem
  • Functions of bounded variation
  • Gal's theorem
  • GCD sums
  • Metric Diophantine Approximation
  • Riemann zeta-function

ASJC Scopus subject areas

  • General Mathematics


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