scholarly journals Randomness and uniform distribution modulo one

2021 ◽  
pp. 104857
Author(s):  
Verónica Becher ◽  
Serge Grigorieff
Keyword(s):  
Uniform Distribution ◽  
Distribution Modulo One ◽  
Uniform Distribution Modulo One ◽  
Distribution Modulo

Uniform Distribution of the Weighted Sum-of-Digits Functions

2021 ◽  
Vol 16 (1) ◽  
pp. 93-126
Author(s):  
Ladislav Mišík ◽  
Štefan Porubský ◽  
Oto Strauch
Keyword(s):  
Uniform Distribution ◽  
Weighted Sum ◽  
Two Dimensional ◽  
Sum Of Digits ◽  
Higher Dimensional ◽  
Distribution Modulo One ◽  
Sum Of Digits Function ◽  
Uniform Distribution Modulo One ◽  
Van Der Corput Sequence ◽  
Distribution Modulo

Abstract The higher-dimensional generalization of the weighted q-adic sum-of-digits functions sq,γ (n), n =0, 1, 2,..., covers several important cases of sequences investigated in the theory of uniformly distributed sequences, e.g., d-dimensional van der Corput-Halton or d-dimensional Kronecker sequences. We prove a necessary and sufficient condition for the higher-dimensional weighted q-adic sum-of-digits functions to be uniformly distributed modulo one in terms of a trigonometric product. As applications of our condition we prove some upper estimates of the extreme discrepancies of such sequences, and that the existence of distribution function g(x)= x implies the uniform distribution modulo one of the weighted q-adic sum-of-digits function sq,γ (n), n = 0, 1, 2,... We also prove the uniform distribution modulo one of related sequences h 1 sq, γ (n)+h 2 sq,γ (n +1), where h 1 and h 2 are integers such that h 1 + h 2 ≠ 0 and that the akin two-dimensional sequence sq,γ (n), sq,γ (n +1) cannot be uniformly distributed modulo one if q ≥ 3. The properties of the two-dimensional sequence sq,γ (n),s q,γ (n +1), n =0, 1, 2,..., will be instrumental in the proofs of the final section, where we show how the growth properties of the sequence of weights influence the distribution of values of the weighted sum-of-digits function which in turn imply a new property of the van der Corput sequence.

scholarly journals On Weyl products and uniform distribution modulo one

2017 ◽  
Vol 185 (3) ◽  
pp. 365-395
Author(s):  
Christoph Aistleitner ◽  
Gerhard Larcher ◽  
Friedrich Pillichshammer ◽  
Sumaia Saad Eddin ◽  
Robert F. Tichy
Keyword(s):  
Uniform Distribution ◽  
Distribution Modulo One ◽  
Uniform Distribution Modulo One ◽  
Distribution Modulo

scholarly journals Notes on uniform distribution modulo one

1990 ◽  
Vol 49 (2) ◽  
pp. 264-272 ◽  
Author(s):  
G. Myerson ◽  
A. D. Pollington
Keyword(s):  
Distribution Function ◽  
Uniform Distribution ◽  
Asymptotic Distribution ◽  
Fixed Integer ◽  
Fixed Set ◽  
Distribution Modulo One ◽  
Asymptotic Distribution Function ◽  
Uniform Distribution Modulo One ◽  
Distribution Modulo

AbstractWe exhibit a sequence (un) which is not uniformly distributed modulo one even though for each fixed integer k ≥ 2 the sequence (kun) is u.d. (mod 1). Within the set of all such sequences, we characterize those with a well-behaved asymptotic distribution function. We exhibit a sequence (un) which is u.d. (mod 1) even though no subsequence of the form (ukn + j) is u.d. (mod 1) for any k ≥ 2. We prove that, if the subsequences (ukn) are u.d. (mod 1) for all squarefree k which are products of primes in a fixed set P, then (un) is u.d. (mod I) if the sum of the reciprocals of the primes in P diverges. We show that this result is the best possible of its type.

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