UNIFORM-DISTRIBUTION OF GENERALIZED POLYNOMIALS

Generalized polynomials form a natural family of functions which are obtained from polynomials by the use of the greatest integer function, addition, and multiplication. Among other results, we show that if the coefficients of a generalized polynomial q(n) are sufficiently independent then the sequence q(n), n = 1, 2, …, is uniformly distributed (mod 1) (see Theorem 3.1). We also show, for example, that the sequence [αn] βn = 1, 2, …, is uniformly distributed (mod 1) if and only if either α2 ∉ Q and β is irrational or α2 ∈ Q and β is rationally independent of 1, α (see Proposition 5.3). © 1993 Academic Press Inc.