Diophantine approximations and almost periodic functions

Abstract In this paper we investigate the asymptotic behaviour of the classical continuous and unbounded almost periodic function in the Lebesgue measure.Using diophantine approximations we show that this function can be estimated by functions of polynomial type and we give the best polynomial estimation.

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On Bohr almost periodicity

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100064434 ◽

1986 ◽

Vol 99 (3) ◽

pp. 489-493 ◽

Cited By ~ 1

Author(s):

Paul Milnes

Keyword(s):

Periodic Function ◽

Present Author ◽

Almost Periodic Functions ◽

Unusual Feature ◽

Almost Periodicity ◽

Almost Periodic Function ◽

Almost Periodic ◽

Periodic Functions ◽

Almost Automorphic

AbstractThe first examples of Bohr almost periodic functions that are not almost periodic were given by T. -S. Wu. Later, the present author showed that Bohr almost periodic functions could be distal (and not almost periodic) and even merely minimal. Here it is proved that all Bohr almost periodic functions are minimal. The proof yields an unusual feature about the orbit of a Bohr almost periodic function, one which does not characterize Bohr almost periodic functions, but can be used to show that a Bohr almost periodic function f that is point distal must be distal or, if f is almost automorphic, it must be almost periodic. Some pathologies of Bohr almost periodic functions are discussed.

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THE CONNECTION BETWEEN PSEUDO ALMOST PERIODIC FUNCTIONS DEFINED ON TIME SCALES AND ON THE REAL LINE

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972717000041 ◽

2017 ◽

Vol 95 (3) ◽

pp. 482-494 ◽

Cited By ~ 1

Author(s):

CHAO-HONG TANG ◽

HONG-XU LI

Keyword(s):

Time Scales ◽

Almost Periodic Functions ◽

Almost Periodic Function ◽

Almost Periodic ◽

Necessary And Sufficient Condition ◽

Periodic Functions ◽

Pseudo Almost Periodic Functions ◽

Necessary And Sufficient ◽

Pseudo Almost Periodic ◽

Uniformly Continuous

A necessary and sufficient condition for a continuous function $g$ to be almost periodic on time scales is the existence of an almost periodic function $f$ on $\mathbb{R}$ such that $f$ is an extension of $g$. Our aim is to study this question for pseudo almost periodic functions. We prove the necessity of the condition for pseudo almost periodic functions. An example is given to show that the sufficiency of the condition does not hold for pseudo almost periodic functions. Nevertheless, the sufficiency is valid for uniformly continuous pseudo almost periodic functions. As applications, we give some results on the connection between the pseudo almost periodic (or almost periodic) solutions of dynamic equations on time scales and of the corresponding differential equations.

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

Journal of Siberian Federal University Mathematics & Physics ◽

10.17516/1997-1397-2019-12-5-571-578 ◽

2019 ◽

pp. 571-578

Author(s):

Wayne M. Lawton

Keyword(s):

Periodic Function ◽

Riemann Hypothesis ◽

Mahler Measure ◽

Trigonometric Polynomials ◽

Almost Periodic Function ◽

Almost Periodic ◽

Periodic Functions ◽

The Mean ◽

Relationship Of ◽

The Relationship

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

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