Discrepancy estimates for generalized polynomials (Q667815)

Let $[y]$ denote the integral part of $y$ and let $\{y\}$ be the fractional part of $y$. Let $\left(x_n\right)_{n\geq 0}$ be a sequence of real numbers and $N$ be any positive integer. The discrepancy of $\left(x_n\right)_{n\geq 0}$, denoted by $D_{N}(x_n)$, is defined by \[ D_{N}(x_n)=\sup_{0\leq a<b\leq 1}\left|\frac{\#\left\{n\leq N: \ \{x_n\}\in[a,b)\right\}}{N}-(b-a)\right|. \] Let $p(x)\in\mathbb{R}[x]$ be a monic polynomial of degree $d\geq 2$, and let $\alpha,\beta$ be non-zero real numbers satisfying certain conditions. The authors obtain an upper bound for the discrepancy of the sequence $\displaystyle \left(\left[p(n)\alpha\right]\beta\right)_{n\geq 0}$.