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

## TL;DR

This paper establishes bounds on the measure of small-value sets of Bohr almost periodic functions with bounded spectrum, linking these bounds to the Mahler measure and the Riemann Hypothesis.

## Contribution

It proves a universal measure estimate for small values of such functions and relates the Mahler measure of their lifts to deep number theory conjectures.

## Key findings

- Mean measure of small-value sets is bounded by a power of u.
- For trigonometric polynomials, the bound depends only on the number of frequencies and largest coefficient.
- Positivity of Mahler measure is established, with implications for the Riemann Hypothesis.

## 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.