Abstract
Let G denote a compact monothetic group, and let \(\rho (x) = \alpha _k x^k + \ldots + \alpha _1 x + \alpha _0\), where \(\alpha _0, \ldots , \alpha _k\) are elements of G one of which is a generator of G. Let \((p_n)_{n\ge 1}\) denote the sequence of rational prime numbers. Suppose \(f \in L^{p}(G)\) for \(p> 1\). It is known that if
then the limit \(\lim _{n\rightarrow \infty } A_Nf(x)\) exists for almost all x with respect Haar measure. We show that if G is connected then the limit is \(\int _{G} f d\lambda \). 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.
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Acknowledgements
The authors thank Buket Eren Gökmen for comments that much improved the readability of the manuscript. Radhakrishnan Nair thanks Laboratoire de Mathématique de l’Université Savoie Mont Blanc for its hospitality and financial support while this paper was being written We also thank the referee for very detailed comments which materially improved the paper.
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Communicated by Karlheinz Gröchenig.
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Hančl, J., Nair, R. & Verger-Gaugry, JL. On polynomials in primes, ergodic averages and monothetic groups. Monatsh Math 204, 47–62 (2024). https://doi.org/10.1007/s00605-024-01948-0
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DOI: https://doi.org/10.1007/s00605-024-01948-0