Abstract
We obtain a sufficient condition for a subsetH of positive integers to satisfy that the equidistribution (mod 1) of the sequences (u n+h − u n; n=1, 2, ···) for allh ∈H implies the equidistribution of (u n). Our condition is satisfied, for example, for the following sets: (1)H={n − m; n ∈ I, m ∈ I, n>m}, whereI is any infinite subset of integers; (2)H={| ψ (n)|; ψ(n)≠0,n ∈ Z}, where ψ is a nonconstant polynomial with integral coefficients having at least one integral zero (modq) for allq=2, 3, ···; (3)H={p+1;p is a prime} andH={p − 1;p is a prime}.
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Kamae, T., Mendes France, M. Van der corput’s difference theorem. Israel J. Math. 31, 335–342 (1978). https://doi.org/10.1007/BF02761498
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DOI: https://doi.org/10.1007/BF02761498