Abstract
Let H be a Hardy field (a field consisting of germs of real-valued functions at infinity that is closed under differentiation) and let f ∈ H be a subpolynomial function. Let P be the sequence of naturally ordered primes. We show that (f(n))n∈ℕ is uniformly distributed mod1 if and only if (f (p))p∈P is uniformly distributed mod 1. This result is then utilized to derive various ergodic and combinatorial statements which significantly generalize the results obtained in [BKMST].
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The first author gratefully acknowledges the support of the NSF under grant DMS-1162073.
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Bergelson, V., Kolesnik, G. & Son, Y. Uniform distribution of subpolynomial functions along primes and applications. JAMA 137, 135–187 (2019). https://doi.org/10.1007/s11854-018-0068-1
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DOI: https://doi.org/10.1007/s11854-018-0068-1