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

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.