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 Chebyshevskii Sb., 2017, Volume 18, Issue 4, Pages 107–115 (Mi cheb600)

Distribution of zeros of nondegenerate functions on short cuttings

V. I. Bernika, N. V. Budarinab, A. V. Lunevicha, H. O'Donnelc

a Institute of Mathematics of the National Academy of Sciences of Belarus
b Dundalk Institute of Technology
c Dublin Institute of Technology

Abstract: 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 $f_1(x), ..., f_n(x)$ 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 $F(x)=a_nf_n(x)+…+ a_1f_1(x)+a_0, a_j\in Z, 1\leq j \leq n$ 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 $F(x)$ on the interval $I$, with $|a_j|\leq Q, 0 \leq\gamma<1$. 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.

DOI: https://doi.org/10.22405/2226-8383-2017-18-4-106-114

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UDC: 511.42
Accepted:14.12.2017

Citation: V. I. Bernik, N. V. Budarina, A. V. Lunevich, H. O'Donnel, “Distribution of zeros of nondegenerate functions on short cuttings”, Chebyshevskii Sb., 18:4 (2017), 107–115

Citation in format AMSBIB
\Bibitem{BerBudLun17} \by V.~I.~Bernik, N.~V.~Budarina, A.~V.~Lunevich, H.~O'Donnel \paper Distribution of zeros of nondegenerate functions on short cuttings \jour Chebyshevskii Sb. \yr 2017 \vol 18 \issue 4 \pages 107--115 \mathnet{http://mi.mathnet.ru/cheb600} \crossref{https://doi.org/10.22405/2226-8383-2017-18-4-106-114}