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- Title
Density of Some Special Sequences Modulo 1.
- Authors
Dubickas, Artūras
- Abstract
In this paper, we explicitly describe all the elements of the sequence of fractional parts { a f (n) / n } , n = 1 , 2 , 3 , ... , where f (x) ∈ Z [ x ] is a nonconstant polynomial with positive leading coefficient and a ≥ 2 is an integer. We also show that each value w = { a f (n) / n } , where n ≥ n f and n f is the least positive integer such that f (n) ≥ n / 2 for every n ≥ n f , is attained by infinitely many terms of this sequence. These results combined with some earlier estimates on the gaps between two elements of a subgroup of the multiplicative group Z m * of the residue ring Z m imply that this sequence is everywhere dense in [ 0 , 1 ] . In the case when f (x) = x this was first established by Cilleruelo et al. by a different method. More generally, we show that the sequence { a f (n) / n d } , n = 1 , 2 , 3 , ... , is everywhere dense in [ 0 , 1 ] if f ∈ Z [ x ] is a nonconstant polynomial with positive leading coefficient and a ≥ 2 , d ≥ 1 are integers such that d has no prime divisors other than those of a. In particular, this implies that for any integers a ≥ 2 and b ≥ 1 the sequence of fractional parts { a n / n b } , n = 1 , 2 , 3 , ... , is everywhere dense in [ 0 , 1 ] .
- Subjects
EULER theorem; INTEGERS
- Publication
Mathematics (2227-7390), 2023, Vol 11, Issue 7, p1727
- ISSN
2227-7390
- Publication type
Article
- DOI
10.3390/math11071727