Distribution of zeros of nondegenerate functions on short cuttings
Čebyševskij sbornik, Tome 18 (2017) no. 4, pp. 107-115
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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.
Keywords:
nondegenerate functionsons, zeros of nondegenerate functionsons.
@article{CHEB_2017_18_4_a3,
author = {V. I. Bernik and N. V. Budarina and A. V. Lunevich and H. O'Donnel},
title = {Distribution of zeros of nondegenerate functions on short cuttings},
journal = {\v{C}eby\v{s}evskij sbornik},
pages = {107--115},
publisher = {mathdoc},
volume = {18},
number = {4},
year = {2017},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/CHEB_2017_18_4_a3/}
}
TY - JOUR AU - V. I. Bernik AU - N. V. Budarina AU - A. V. Lunevich AU - H. O'Donnel TI - Distribution of zeros of nondegenerate functions on short cuttings JO - Čebyševskij sbornik PY - 2017 SP - 107 EP - 115 VL - 18 IS - 4 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/CHEB_2017_18_4_a3/ LA - ru ID - CHEB_2017_18_4_a3 ER -
%0 Journal Article %A V. I. Bernik %A N. V. Budarina %A A. V. Lunevich %A H. O'Donnel %T Distribution of zeros of nondegenerate functions on short cuttings %J Čebyševskij sbornik %D 2017 %P 107-115 %V 18 %N 4 %I mathdoc %U http://geodesic.mathdoc.fr/item/CHEB_2017_18_4_a3/ %G ru %F CHEB_2017_18_4_a3
V. I. Bernik; N. V. Budarina; A. V. Lunevich; H. O'Donnel. Distribution of zeros of nondegenerate functions on short cuttings. Čebyševskij sbornik, Tome 18 (2017) no. 4, pp. 107-115. http://geodesic.mathdoc.fr/item/CHEB_2017_18_4_a3/