Distribution of Small Values of Bohr Almost Periodic Functions with Bounded Spectrum

Abstract

If $f$ is a nonzero Bohr almost periodic function on $\mathbb{R}$ with a bounded spectrum we prove 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 $\le n+1$ frequencies we show that $C_f$ can be chosen to depend only on $n$ and the modulus of the largest coefficient of $f.$ We show 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 discuss the relationship of Mahler measure to the Riemann Hypothesis.