Pseudorandom numbers and entropy conditions

We investigate measures of pseudorandomness of finite sequences (xn) of real numbers. Mauduit and Sárközy introduced the “well-distribution measure”, depending on the behavior of the sequence (xn) along arithmetic subsequences (xak+b). We extend this definition by replacing the class of arithmetic progressions by an arbitrary class A of sequences of positive integers and show that the so obtained measure is closely related to the metric entropy of the class A. Using standard probabilistic techniques, this fact enables us to give precise bounds for the pseudorandomness measure of classical constructions. In particular, we will be interested in “truly” random sequences and sequences of the form {nkω}, where {·} denotes fractional part, (nk) is a given sequence of integers and ω∈[0,1).