On series $\sum c_k f(kx)$ and Khinchin's conjecture

TL;DR

This paper proves the optimality of a criterion related to Khinchin's conjecture, resolving a long-standing open problem, and provides near optimal conditions for almost everywhere convergence of certain series involving functions in L^2.

Contribution

It establishes the optimality of Koksma's criterion in Khinchin's conjecture and offers near optimal convergence conditions for series of the form  rac{c_k}{f(kx)} for functions in L^2.

Findings

Proved the optimality of Koksma's criterion in Khinchin's conjecture.

Resolved a long-standing open problem in analysis.

Provided near optimal conditions for a.e. convergence of series  c_k f(kx).

Abstract

We prove the optimality of a criterion of Koksma (1953) in Khinchin's conjecture, settling a long standing open problem in analysis. Using this result, we also give a near optimal condition for the a.e.\ convergence of series $\sum_{k=1}^\infty c_k f(kx)$ for $f\in L^2$.