Integer polynomials and Minkowski's theorem on linear forms

V. I. Bernik ; I. A. Korlyukova ; A. S. Kudin ; A. V. Titova

Čebyševskij sbornik, Tome 23 (2022) no. 1, pp. 45-52

Résumé

In paper Minkowski's theorem on linear forms [1] is applied to polynomials with integer coefficients \begin{align} P(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \end{align} with degree $degP = n$ and height $H(P)=\max_{0 \le i \le n} |a_i|$. Then, for any $x \in [0,1)$ and a natural number $Q > 1$, we obtain the inequality \begin{align} |P(x)| c_1(n) Q ^{-n} \end{align} for some $P(x), H(P) \leq Q$. Inequality (4) means that the entire interval $[0,1)$ can be covered by intervals $I_i, i = 1, 2, \ldots$ at all points of which inequality (4) is true. An answer is given to the question about the size of the $I_i$ intervals. The main result of this paper is proof of the following statement. For any $v$, $0 \leq v \frac{n+1}{3}$, there is an interval $J_k$, $k = 1, \ldots, K$, such that for all $x \in J_k$, the inequality (4) holds and, moreover, \begin{align*} c_2 Q^{-n-1+v} \mu J_k c_3 Q^{-n-1+v}. \end{align*}

Détail

Keywords: diophantine approximation, Lebesgue measure, Minkowski's theorem.