РАСПРЕДЕЛЕНИЕ НУЛЕЙ НЕВЫРОЖДЕННЫХ ФУНКЦИЙ НА КОРОТКИХ ОТРЕЗКАХ

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 f1 (x), ..., fn (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) = anfn (x) + ... + a1f1 (x) + a0, aj ∈ Z, 1 ≤ j ≤ 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−γ, 0 ≤ γ < 1. We obtain upper and lower bounds for the number of zeros of the function F (x) on the interval I, with |aj| ≤ Q, 0 ≤ γ < 1. For γ = 0 such estimates were obtained by A. S. Pyartli, V. G. Sprindzhuk, V. I. Bernik, V. V. Beresnevitch, N. V. Budarina.