Sums with Stern-Brocot sequences and Minkowski question mark function

Abstract

We give an affirmative answer to a question asked by N. Moshchevitin \cite{m1} in his lecture at International Congress of Basic Science, Beijing, 2024 (see also \cite{m}, Section 6.3). The question is that whether the remainder $$ R_n=\sum_{j=1}^{2^n}\left(\xi_{j,n}-\frac{j}{2^n}\right)^2-2^n\int_0^1( ?(x)-x))^2\text{d}x $$ tends to $0$ when $n$ tends to infinity, where $\xi_{j,n}$ are elements of the Stern-Brocot sequence and $?(x)$ denotes Minkowski Question-Mark Function. We present some extended results and give a correct proof of a theorem on the Fourier-Stieltjes coefficient of the inverse function of $?(x)$.