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

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 $f$.