Geodesic
Distribution of small values of Bohr almost periodic functions with bounded spectrum
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 12 (2019) no. 5, pp. 571-578

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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.
Keywords: almost periodic function, entire function, Beurling factorization, Mahler measure, Riemann hypothesis. 
Wayne M. Lawton. Distribution of small values of Bohr almost periodic functions with bounded spectrum. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 12 (2019) no. 5, pp. 571-578. http://geodesic.mathdoc.fr/item/JSFU_2019_12_5_a4/
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