On the distribution of fractional parts of polynomials

O. M. Fomenko

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 26, Tome 392 (2011), pp. 191-201

Résumé

In the paper upper bounds for sums of the form $$ \sum_{0\le N}\psi\big(f(n)\big), $$ where $f(x)$ is a polynomial and $\psi(x)=x-[x]-1/2$, are obtained. The cases $$ f(x)=\frac1\alpha x^2+\beta x+\gamma $$ and $$ f(x)=\frac1\alpha x^3+\beta x^2+\gamma x+\delta $$ are considered, where $\alpha$ is a large positive number. Weyl's method and V. N. Popov's reasoning (Mat. Zametki, 18 (1975), 699–704) are used.