On some Rajchman measures and equivalent Salem's problem

TL;DR

This paper constructs specific Rajchman measures using Fourier transform properties and establishes their connection to Salem's longstanding open problem about the decay of Fourier-Stieltjes coefficients of the Minkowski question mark function.

Contribution

It introduces a new approach to construct Rajchman measures and links an open problem in harmonic analysis to the decay of Fourier-Stieltjes coefficients of a special function.

Findings

Constructed new Rajchman measures using Fourier analysis.

Proved the equivalence of a problem on measure decay to Salem's open question.

Connected Fourier-Stieltjes coefficient decay to Minkowski's question mark function.

Abstract

We construct certain Rajchman measures by using integrability properties of the Fourier and Fourier-Stieltjes transforms. In particular, we state a problem and prove that it is equivalent to the known and still unsolved question posed by R. Salem (Trans. Amer. Math. Soc. 53 (3) (1943), p. 439) whether Fourier-Stieltjes coefficients of the Minkowski's question mark function vanish at infinity.