On the fractional parts of lacunary sequences

Autores: A. Dubickas
Localización: Mathematica scandinavica, ISSN 0025-5521, Vol. 99, Nº 1, 2006, págs. 136-146
Idioma: inglés
DOI: 10.7146/math.scand.a-15004
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Resumen
- In this paper, we prove that if t0,t1,t2,… is a lacunary sequence, namely, tn+1/tn≥1+r−1 for each n≥0, where r is a fixed positive number, then there are two positive constants c(r)=max(1−r,2(3r+6)−2) and ξ=ξ(t0,t1,…) such that the fractional parts {ξtn}, n=0,1,2,…, all belong to a subinterval of [0,1) of length 1−c(r). Some applications of this theorem to the chromatic numbers of certain graphs and to some fast growing sequences are discussed. We prove, for instance, that the number 10−−√ can be written as a quotient of two positive numbers whose decimal expansions contain the digits 0, 1, 2, 3 and 4 only.