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 $\leq n + 1$ 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.