Recently, Cilleruelo, Kumchev, Luca, Rué and Shparlinski proved that for each integer a ≥ 2 the sequence of fractional parts is everywhere dense in the interval [0,1]. We prove a similar result for all Pisot numbers and Salem numbers α and show that for each c > 0 and each sufficiently large N, every subinterval of [0,1] of length contains at least one fractional part Q(αⁿ)/n, where Q is a nonconstant polynomial in ℤ[z] and n is an integer satisfying 1 ≤ n ≤ N.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm128-2-9,
author = {Art\=uras Dubickas},
title = {Density of some sequences modulo 1},
journal = {Colloquium Mathematicae},
volume = {126},
year = {2012},
pages = {237-244},
zbl = {1316.11066},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm128-2-9}
}
Artūras Dubickas. Density of some sequences modulo 1. Colloquium Mathematicae, Tome 126 (2012) pp. 237-244. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm128-2-9/