In this paper we seak to study the discrepancy and consequently the uniform distribution mod 1 of two types of sequences. For this purpose we denote by [ ] the integer part function. For real c with 1<c< 3 2 take (![n c ] : n=1, 2, 3, ...); here the real ! is badly approximable in the sense of *-admissibility (Theorem 1). For real c, q with 1<c< 7 6 , 0<q<1, take also ([n c ](log n) q : n=1, 2, 3, ...) (Theorem 3).
1997 Academic Press
Recently, Ha# land [2] proved interesting results concerning the integer part function [ ] and generalized polynomials. This prompted me to look at [ ] and uniform distribution mod 1. Section 1 contains a criterion of for numbers of the form [n c ]. In Section 2 we prove an upper estimate for the discrepancy of the sequences (![n c ] : n=1, 2, 3, ...) with badly approximable ! (Theorem 1). Section 3 has been inserted for completeness only. In Section 4 we prove an upper estimate for the discrepancy of the sequences ([n c ](log n) q : n=1, 2, 3, ...) for real c, q with 1<c< 7 6 , 0<q<1 (Theorem 3). For the proofs we use results of Erdo s and Tura n [1] on the discrepancy, of Vinogradov [9] on Fourier series, and of van der Corput (see [8, p. 90]) on exponential sums (using the second and the third derivatives).