Salem's problem for the inverse Minkowski $?(t)$ function

E. P. Golubeva

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 29, Tome 429 (2014), pp. 11-19

Résumé

Let $d_n$ be the coefficient Fourier–Stieltjes of the Minkowski $?(t)$ function – $$ d_n=\int^1_0\cos2\pi nt\,d?(t). $$ Salem's problem is as to whether $d_n$ tends to zero as $n\to\infty$. In the paper the coefficient Fourier $$ \alpha_n=\int^1_0\cos(2\pi n?(t))\,dt $$ is considered. It is proved that $\alpha_n$ does not tend to zero as $n\to\infty$.