An Extension of Weyl’s Equidistribution Theorem to Generalized Polynomials and Applications

DOI10.1093/IMRN/RNAA035zbMATH Open1483.11153arXiv1911.05938OpenAlexW3009786213MaRDI QIDQ5021584

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Publication date: 13 January 2022

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Abstract: Generalized polynomials are mappings obtained from the conventional polynomials by the use of operations of addition, multiplication and taking the integer part. Extending the classical theorem of H. Weyl on equidistribution of polynomials, we show that a generalized polynomial

q (n)

has the property that the sequence

(q (n) l a m b d a)_{n i n m a t h b b Z}

is well distributed

for all but countably many

l a m b d a i n m a t h b b R

if and only if

l i m l i m i t s_{s u b s t a c k | n | i g h t a r r o w i n f t y n o t i n J} | q (n) | = i n f t y

for some (possibly empty) set

J

having zero density in

m a t h b b Z

. We also prove a version of this theorem along the primes (which may be viewed as an extension of classical results of I. Vinogradov and G. Rhin). Finally, we utilize these results to obtain new examples of sets of recurrence and van der Corput sets.

Full work available at URL: https://arxiv.org/abs/1911.05938

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