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

Abstract

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 f x : jf(x)j < u g is less than Cf u1=n: For trigonometric polynomials 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; is positive and discussed the relationship of Mahler measure to the Riemann Hypothesis