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Certain averages on the $ {\bf a}$-adic numbers

Authors: Nakhlé H. Asmar and Radhakrishnan Nair
Journal: Proc. Amer. Math. Soc. 114 (1992), 21-28
MSC: Primary 11K41; Secondary 11K55, 22D40, 28D99, 47A35
MathSciNet review: 1087460
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Abstract: For $ {L^p} \cap {L^2}$ functions $ f$, with $ p$ greater than one, defined on the $ {\mathbf{a}}$-adic numbers $ {\Omega _{\mathbf{a}}}$, we consider averages like

$\displaystyle A_N^{\left( 1 \right)}f\left( {\mathbf{x}} \right) = \frac{1}{N}\... ...{1}{N}\sum\limits_{n = 1}^N {f\left( {{\mathbf{x}} + {p_n}\alpha } \right),} } $

where $ {\mathbf{x}}$ and $ \alpha $ are in $ {\Omega _{\mathbf{a}}}$. Here $ {p_n}$ denotes the $ n$th prime. These averages are known to converge for almost all $ {\mathbf{x}}$. We describe explicitly these limits, which possibly contrary to expectation, turn out in general not to be the integral of $ f$.

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Article copyright: © Copyright 1992 American Mathematical Society

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