On Monotone Almost (P, μ) - u. d. Mod 1 Sequence
✍ Scribed by Kazuo Goto
- Publisher
- John Wiley and Sons
- Year
- 2010
- Tongue
- English
- Weight
- 435 KB
- Volume
- 205
- Category
- Article
- ISSN
- 0025-584X
No coin nor oath required. For personal study only.
✦ Synopsis
Using the result of almost u.d. mod 1 and G. HOHEISEL [6] and Emos's result, AKIYAMA [l] gave a different proof of the ERDOS and T U ~N result [4], i.e., A210gpn changes its sign infinitely many times, where pn is the n-th prime number and Aaf(n
In this paper, we show the neceasary condition for monotone almost uniform distribution mod 1. We give FejBr's type theorem for almost uniform distribution mod 1 and give a similar result of NIEDERREITER [a]. Also we obtain the converse of its result with some conditions. As an application, we give a simple proof [l] of ERDOS and TURAN'S result using ERD~S'S result [cf. 3, 9, 10, 11, and 21. Also we obtain the condition of P ( n ) that Aa log P(n) changes its sign infinitely many times.
f(n + 1)f(n).
1. Results
Let (p(n)) be a non-negative sequence with p(1) > 0 and s(n) = p ( l ) + . . . + p ( n ) -+ co as n + 00. Let p(z) be a regular Bore1 probability measure that is not a point measure, {z} is the fractional part of z and C([O,t);g(j)) is 1 if { g ( j ) } E [ 0 7 t ) , 0 otherwise. Definition 1.1. Let (g(n)),",l be a sequence of real numbers. If there exists an increasing sequence of natural numbers (A) = {XI, XZ, , . . } which tends to infinity such that for all t E (0, l), then the sequence (g(n)) is called X -almost (p, p ) -u. d. mod 1 and p ( z ) is called a X -almost pdistribution function mod 1 (d. f. mod 1) of (g(n)). If p(z) = z, then (g(n)) is called Xalmost p -u. d. mod 1.