Estimation of the mean value of the remainder term in the asymptotic formula for the sum of values of an arithmetical function on a Beatty sequence

The paper is concerned with the estimation of average values of $\Delta(\alpha,N)=\Delta(\alpha,0,N)$ and $\Delta(\alpha,\beta,N)$ with respect to $\alpha>1$ and $0<\beta<\alpha$ respectively, where $\Delta(\alpha,\beta,N)$ denotes the remainder term in the formula of the form
$$\sum_{n\leq N}f([\alpha n+\beta])=\frac{1}{\alpha}\sum_{m\leq \alpha N+\beta}f(m)+\Delta(\alpha,\beta,N),$$
for an arbitrary number-theoretical fuction $f(n)$.