Acta Math. Hungar. 66 (1-2) (1995), 113-126. NUMBERS WITH COMPLICATED DECIMAL EXPANSIONS D. BEREND (Beer-Sheva) and M. D. BOSHERNITZAN 1 (Houston) 1. Introduction One of the most basic results in the theory of distribution modulo 1 is that, if a is an irrational, then the sequence (na)~=l is dense, and even uniformly distributed, modulo 1. In particular, given any digits al, as,..., ak, there exists a positive integer m for which the decimal expansion of ma contains this block of digits. A considerable strengthening of this result was obtained by Mahler [13] who proved that, moreover, there necessarily exists an m for which the decimal expansion of ma contains the given block infinitely often. Mahler also established an upper bound for the minimal value M of the number m with that property; M = M(k) depends only on the number k of digits, but not on a. It was noted by Furstenberg that, employing a certain result of Glasner [10], one can provide a very short proof of the finiteness of M(k) (see [1, Corollary 7.2]). Motivated by his approach, the authors [4] gave another short proof of Mahler's result, which at the same time yielded a better upper bound, best possible up to a constant factor. The density modulo 1 of the sequence (ha) is a special case of a result which asserts that, given any polynomial P with real coefficients, at least one of which (besides the free term) is irrational, the sequence P(n) is dense modulo 1. (Better known is Weyl's even stronger result by which this sequence is uniformly distributed modulo 1 [15].) It was shown in [4, Theorem 1.2] that Mahler's result is true in this more general setting as well. A few other sequences besides polynomial sequences, for example (log n) and (n ~ for positive rational non-integer 8, were shown to satisfy the same property. In this paper we present a general framework for the study of sequences in which there exist terms whose expansions tend to be complicated in the sense that they contain "numerous" blocks, perhaps appearing "many" times. In Section 2 we introduce the required definitions and show that some sequences, and families of sequences, possess these properties. Section 3 deals with 1 Research supported in part by NSF Grant No. DMS-9003450. 0236-5294/95/$4.00 9 1995 Akad~miai Kiadb, Budapest