FUNCTIONS ON SHORT CUTTINGS
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),..., (𝑥) be functions differentiable on the interval 𝐼, +1 times and Wronskian from derivatives almost everywhere on is different from 0. Such functions are called nondegenerate. The problem of the distribution of the zeros of the function (𝑥)= (𝑥)+... + 𝑎1𝑓1 (𝑥)+ 𝑎0, ∈ 𝑍, 1 ≤ ≤ is important in the metric theory of Diophantine approximations. Let > 1 be a sufficiently large integer, and the interval has length 𝑄-𝛾  , 0 ≤ < 1. We obtain upper and lower bounds for the number of zeros of the function (𝑥) on the interval 𝐼, with |𝑎𝑗|≤ 𝑄, 0 ≤ < 1. For =0 such estimates were obtained by A. S. Pyartli, V. G. Sprindzhuk, V. I. Bernik, V. V. Beresnevitch, N. V. Budarina.