Gaps in the sequence n 2 ϑ ( mod 1 )

Vladimir Drobot

International Journal of Mathematics and Mathematical Sciences, 1987, vol. 10, 1-4

Abstract:

Let ϑ be an irrational number and let { t } denote the fractional part of t . For each N let I 0 , I 1 , … , I N be the intervals resulting from the partition of [ 0 , 1 ] by the points { k 2 ϑ } , k = 1 , 2 , … , N . Let T ( N ) be the number of distinct lengths these intervals can assume. It is shown that T ( N ) → ∞ . This is in contrast to the case of the sequence { n ϑ } , where T ( N ) ≤ 3 .

Date: 1987
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:347845

DOI: 10.1155/S0161171287000164

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