Distribution of zeros of nondegenerate functions on short cuttings

V. I. Bernik ; N. V. Budarina ; A. V. Lunevich ; H. O'Donnel

Čebyševskij sbornik, Tome 18 (2017) no. 4, pp. 107-115

Résumé

The paper presents newly obtained upper and lower bounds for the number of zeros for functions of a special type, as well as an estimate for the measure of the set where these functions attain small values. Let $f_1\left(x\right), \ ..., \ f_n\left(x\right)$ be functions differentiable on the interval $I$, $n+1$ times and Wronskian from derivatives almost everywhere on $I$ is different from 0. Such functions are called nondegenerate. The problem of the distribution of the zeros of the function $F\left(x\right)=a_nf_n\left(x\right)+\dots+ a_1f_1\left(x\right)+a_0, \ a_j\in Z, \ 1\leq j \leq n$ is important in the metric theory of Diophantine approximations. Let $Q>1$ be a sufficiently large integer, and the interval $I$ has length $Q^{-\gamma}, \ 0\leq \gamma 1$. We obtain upper and lower bounds for the number of zeros of the function $F\left(x\right)$ on the interval $I$, with $\left|a_j\right|\leq Q, \ 0 \leq\gamma1$. For $\gamma=0$ such estimates were obtained by A. S. Pyartli, V. G. Sprindzhuk, V. I. Bernik, V. V. Beresnevitch, N. V. Budarina.

Détail

Keywords: nondegenerate functionsons, zeros of nondegenerate functionsons.