Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 523.10023
Autor: Erdös, Paul; Sarkoezy, A.
Title: On almost divisibility properties of sequences of integers. I. (In English)
Source: Acta Math. Hung. 41, 309-324 (1983).
Review: Let |\alpha| denote the distance from the real number \alpha to the nearest integer. we say, the positive real number b is \epsilon-divisible by the positive real number a (and we then write a|\epsilonb), if |b/a| < \epsilon. The following theorem will be proved. Let \epsilon > 0, n > n1(\epsilon) a positive integer, t a real number such that n < t \leq \exp(log5/4n/ log log n). Let further | k = | [2 log t/ log n]-3 if 2 \leq log t/ log n < c1, |
| [ log t/ log n+0.5] if log t/ log n \geq c1, |
where c1 is a certain positive absolute constant. With | F = | n if n < t < n2 |
| [n1- ½^{k+2}] if n2 \leq t < nc1 |
| [(nk+5/2/t)1/(k+2)] if t \geq nc1 |
there exists a positive integer j such that 1 \leq j \leq P and (n+j)|\epsilont.
The main tool are estimates of certain trigonometric sums. On the other hand it will be shown, that for 0 < \epsilon < 1/4, \delta > 0 and n > n2(\epsilon) there exists a real number t such that n < t < \exp((2+\delta)n) and there does not exist an integer j satisfying l \leq j \leq n and (n+j)|\epsilont. This leads to the inequality \exp(log5/4n/ log log n) \leq f(n,\epsilon) \leq \exp((2+\delta)n) (n \geq n0(\epsilon)) where f(n,\epsilon) denotes the infimum of the real numbers t > n such that for any integer l \leq j \leq n t is not \epsilon-divisible by n+j.
Reviewer: D.Leitmann
Classif.: * 11L03 Trigonometric and exponential sums, general
11A05 Multiplicative structure of the integers
Keywords: almost divisibility; sequences of integers; epsilon-divisibility; estimates of trigonometric sums
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