Distribution of small values of Bohr almost periodic functions with bounded spectrum

Wayne M. Lawton

Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 12 (2019) no. 5, pp. 571-578

Résumé

For $f$ a nonzero Bohr almost periodic function on $\mathbb R$ with a bounded spectrum we proved there exist $C_f > 0$ and integer $n > 0$ such that for every $u > 0$ the mean measure of the set $\{\, x \, : \, |f(x)| u \, \}$ is less than $C_f\, u^{1/n}.$ For trigonometric polynomials with $\leq n + 1$ frequencies we showed that $C_f$ can be chosen to depend only on $n$ and the modulus of the largest coefficient of $f.$ We showed this bound implies that the Mahler measure $M(h),$ of the lift $h$ of $f$ to a compactification $G$ of $\mathbb R,$ is positive and discussed the relationship of Mahler measure to the Riemann Hypothesis.

Détail

Keywords: almost periodic function, entire function, Beurling factorization, Mahler measure, Riemann hypothesis.