𝔖 Bobbio Scriptorium
✦   LIBER   ✦

s(N) - Uniform Distribution Modulo 1

✍ Scribed by M. Drmota; R. Winkler

Publisher
Elsevier Science
Year
1995
Tongue
English
Weight
365 KB
Volume
50
Category
Article
ISSN
0022-314X

No coin nor oath required. For personal study only.

✦ Synopsis

In this paper we study the notion of (s(N))-uniform distribution of sequences modulo 1 which sharpens resp. quantifies the notion of complete uniform distribution. A trivial necessary condition for the existence of (s(N))-u.d. sequences is (s(N)=o(N)). On the other hand (s(N)=o(\sqrt{N / \log N})) implies that almost all sequences are (s(N))-u.d. (It seems to be very difficult to fill the gap between those two results.) By a modification of the metric methods it is shown how to construct an (s(N))-u.d. sequence if (s(N)=o(\sqrt{N / \log N})). This construction is rather involved. So a final section discusses some possibilities of simpler approaches for constructing (s(N))-u.d. sequences. 1995 Academic Press, Inc.

πŸ“œ SIMILAR VOLUMES

The Sum-of-Digits-Function and Uniform D
The Sum-of-Digits-Function and Uniform Distribution Modulo 1
✍ Michael Drmota; Gerhard Larcher πŸ“‚ Article πŸ“… 2001 πŸ› Elsevier Science 🌐 English βš– 210 KB

The aim of this paper is to provide detailed estimates for the discrepancy of the sequences ([: } s q (n)]) ([x] denotes the fractional part of x) and results concerning the uniform distribution and the discrepancy of the sequences ([: 1 } s q 1 (n)], ..., [: d } s q d (n)]), where :, : 1 , ..., : d

On an Inequality of ErdΕ‘s and TurΓ‘n Conc
On an Inequality of ErdΕ‘s and TurΓ‘n Concerning Uniform Distribution Modulo One, II
✍ I.Z. Ruzsa πŸ“‚ Article πŸ“… 1994 πŸ› Elsevier Science 🌐 English βš– 126 KB

A famous inequality of ErdΓΆs and TurΓ‘n estimates the discrepancy \(\Delta\) of a finite sequence of real numbers by the quantity \(B=\min _{K} K^{-1}+\sum_{k=1}^{K-1}\left|\alpha_{k}\right| / k\), where the \(\alpha_{k}\) are the Fourier coefficients. We investigate how bad this estimate can be. We