Distribution of Small Values of Bohr Almost Periodic Functions with Bounded Spectrum
For f a nonzero Bohr almost periodic function on R with a bounded spectrum we proved 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 u <sup>1/n</sup>. For trigonometricpolynomials with : n +1 frequencies we showed that Cf 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 R, ispositive and discussed the relationship of Mahler measure to the Riemann Hypothesis.