An estimate of the free term of a~non-negative trigonometric polynomial with integer coefficients

A. S. Belov ; S. V. Konyagin

Izvestiya. Mathematics , Tome 60 (1996) no. 6, pp. 1123-1182

Résumé

We denote by $M_Z^{\downarrow}(n)$ (resp., $K_Z^{\downarrow}(n)$) the smallest value of $a_0$ that can occur in a non-negative trigonometric polynomial $$ \sum_{k=0}^n a_k\cos(kx) $$ with non-negative integer coefficients $a_1\geqslant a_2\geqslant\dots\geqslant a_n$ such that $a_n\geqslant 1$ (resp., $\sum_{k=1}^n a_k=n$). We prove that for all natural numbers $n\geqslant 3$ $$ \dfrac{\ln^2 n}{\ln\ln n}\ll K_Z^\downarrow(n)\ll M_Z^\downarrow(n)\ll(\ln n)^3. $$