On a problem by R. Salem concerning Minkowski's question mark function

TL;DR

This paper discusses the decay of the Fourier transform of the measure defined by Minkowski's question mark function, highlighting recent results that show polynomial decay speed, thus addressing a problem posed by Salem in 1943.

Contribution

It connects recent advances by Jordan and Sahlsten to Salem's longstanding problem about Fourier decay of the question mark measure.

Findings

Fourier transform of the measure decays to zero

Decay occurs at a polynomial speed

Addresses a problem from 1943

Abstract

Minkowski's question mark function is strictly increasing on $[0, 1]$ and hence defines a Stieltjes measure on $[0, 1]$. A problem originating from Salem in 1943, is to determine whether the Fourier series of this measure decay to zero or not. The purpose of this note is to mention that a recent result by Jordan and Sahlsten implies that the Fourier transform decays to zero with a polynomial speed.