The Minkowski ?(x) function and Salem's problem

TL;DR

This paper explores Salem's problem regarding the Fourier-Stieltjes transform of the Minkowski question mark function, proposing approaches and analyzing related functional equations, ultimately supporting an affirmative answer and critiquing recent false attempts.

Contribution

It introduces new approaches to Salem's problem by analyzing the functional equations satisfied by the transform and critiques recent incorrect solutions.

Findings

Transform satisfies integral and discrete functional equations

Supports the affirmative answer to Salem's problem

Recent attempt to solve the problem is fallacious

Abstract

R. Salem (Trans. Amer. Math. Soc. 53 (3) (1943) 427-439) asked whether the Fourier-Stieltjes transform of the Minkowski question mark function ?(x) vanishes at infinity. In this note we present several possible approaches towards the solution. For example, we show that this transform satisfies integral and discrete functional equations. Thus, we expect the affirmative answer to Salem's problem. In the end of this note we show that recent attempt to settle this question (S. Yakubovich, C. R. Acad. Sci. Paris, Ser. I 349 (11-12) (2011) 633-636) is fallacious.