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

Authors:
NakhlĂ© H. Asmar and Radhakrishnan Nair

Journal:
Proc. Amer. Math. Soc. **114** (1992), 21-28

MSC:
Primary 11K41; Secondary 11K55, 22D40, 28D99, 47A35

DOI:
https://doi.org/10.1090/S0002-9939-1992-1087460-6

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 \[ A_N^{\left ( 1 \right )}f\left ( {\mathbf {x}} \right ) = \frac {1}{N}\sum \limits _{n = 1}^N {f\left ( {{\mathbf {x}} + {n^2}\alpha } \right ),{\text { and }}A_N^{\left ( 2 \right )}f\left ( {\mathbf {x}} \right ) = \frac {1}{N}\sum \limits _{n = 1}^N {f\left ( {{\mathbf {x}} + {p_n}\alpha } \right ),} } \] 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