2019/04/19 by Wayne Lawton, Lawton, Wayne
Mathematics · #11K70 #11R06 #30D15 #47A68 #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Approximation and Integration #Meromorphic and Entire Functions #advanced mathematical theories #math.FA #msc:11K70 #msc:11R06 #msc:30D15 #msc:47A68
paper · pdf · doi:10.48550/arxiv.1904.09373
arxiv created 2019/04/19 · openalex publication_date 2019/04/19 · arxiv updated 2019/04/23 · openalex created_date 2022/07/29 · openalex updated_date 2026/07/28
If f is a nonzero Bohr almost periodic function on \mathbb R with a bounded spectrum we prove there exist Cf > 0 and integer n > 0 such that for every u > 0 the mean measure of the set \ x : |f(x)| < u \ is less than Cf u1/n. For trigonometric polynomials with ≤ n + 1 frequencies we show that Cf 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.