On the fractional parts of lacunary sequences - Dubickas, Arturas (MATHEMATICA SCANDINAVICA, 2006)
In this paper, we prove that if t_0, t_1, t_2, … is a lacunary sequence, namely, t_n+1/t_n≥ 1+r^-1 for each n≥ 0, where r is a fixed positive number, then there are two positive constants c(r)=(1-r, 2(3r+6)^-2) and ξ=ξ(t_0, t_1,…) such that the fractional parts {ξ t_n}, 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.