On the intersection of two homogeneous Beatty sequences

Homogeneous Beatty sequences are sequences of the form $a_n=[\alpha n]$, where $\alpha$ is a positive irrational number. In 1957 T. Skolem showed that if the numbers $1,\frac{1}{\alpha},\frac{1}{\beta}$ are linearly independent over the field of rational numbers, then the sequences $[\alpha n]$ and $[\beta n]$ have infinitely many elements in common. T. Bang strengthened this result: denote $S_{\alpha,\beta}(N)$ the number of natural numbers $k$, $1\leqslant k\leqslant N$, that belong to both Beatty sequences $[\alpha n]$, $[\beta m]$, and the numbers $1,\frac{1}{\alpha},\frac{1}{\beta}$ are linearly independent over the field of rational numbers, then $S_{\alpha,\beta}(N)\sim \frac{N}{\alpha\beta}$ for $N\to\infty.$
In this paper, we prove a refinement of this result for the case of algebraic numbers. Let $\alpha,\beta>1$ be irrational algebraic numbers such that $1,\frac{1}{\alpha},\frac{1}{\beta}$ are linearly independent over the field of rational numbers. Then for any $\varepsilon>0$ the following asymptotic formula holds:
$$S_{\alpha,\beta}(N)=\frac{N}{\alpha\beta}+O\bigl(N^{\frac12+\varepsilon}\bigr), N\to\infty.$$