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 $$R_n=\sum _{j=1}^{2^n}{\left(\hat{I}{¾}_{j,n}-\frac{j}{2^n}\right)}^2-2^n{\int }_0^1(?(x)-x){)}^2\text{d}x$$ tends to $0$ when $n$ tends to infinity, where $\hat{I}{¾}_{j,n}$ 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 $?(x)$.