On polynomials in primes, ergodic averages and monothetic groups

Hančl, Jaroslav, Nair, Radhakrishnan and Verger-Gaugry, Jean-Louis (2024) On polynomials in primes, ergodic averages and monothetic groups. Monatshefte für Mathematik, 204 (1). pp. 47-62.

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Abstract

Let G denote a compact monothetic group, and let ρ(x)=αkxk+…+α1x+α0, where α0,…,αk are elements of G one of which is a generator of G. Let (pn)n≥1 denote the sequence of rational prime numbers. Suppose f∈Lp(G) for p>1. It is known that if (Formula presented.) then the limit limn→∞ANf(x) exists for almost all x with respect Haar measure. We show that if G is connected then the limit is ∫Gfdλ. In the case where G is the a-adic integers, which is a totally disconnected group, the limit is described in terms of Fourier multipliers which are generalizations of Gauss sums.

Item Type:	Article
Divisions:	Faculty of Science and Engineering > School of Physical Sciences
Depositing User:	Symplectic Admin
Date Deposited:	20 Mar 2024 10:27
Last Modified:	26 Apr 2024 02:25
DOI:	10.1007/s00605-024-01948-0
Related URLs:	Publisher
URI:	https://livrepository.liverpool.ac.uk/id/eprint/3179457