Distribution functions of ratio sequences, IV

Abstract

In this paper we continue our study of distribution functions g(x) of the sequence of blocks $X_n = (\tfrac{{x_1 }} {{x_n }},\tfrac{{x_2 }} {{x_n }},...,\tfrac{{x_n }} {{x_n }}) $ , n = 1, 2, …, where x n is an increasing sequence of positive integers. Applying a special algorithm we find a lower bound of g(x) also for x n with lower asymptotic density $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{d} $ = 0. This extends the lower bound of g(x) for x n with $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{d} $ > 0 found in the previous part III. We also prove that for an arbitrary real sequence y n ∈ [0, 1] there exists an increasing sequence xn of positive integers such that any distribution function of y n is also a distribution function of X n .