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Keywords:
uniform distribution modulo $1$; equidistribution in probability; algebraic number fields; $S$-adele ring; $S$-integer dynamical system; algebraic dynamics; topological dynamics; $a$-adic solenoid
uniform distribution modulo $1$; equidistribution in probability; algebraic number fields; $S$-adele ring; $S$-integer dynamical system; algebraic dynamics; topological dynamics; $a$-adic solenoid
Summary:
Let $X$ be the quotient group of the $S$-adele ring of an algebraic number field by the discrete group of $S$-integers. Given a probability measure $\mu $ on $X^d$ and an endomorphism $T$ of $X^d$, we consider the relation between uniform distribution of the sequence $T^n\bold {x}$ for $\mu $-almost all $\bold {x}\in X^d$ and the behavior of $\mu $ relative to the translations by some rational subgroups of $X^d$. The main result of this note is an extension of the corresponding result for the $d$-dimensional torus $\mathbb T^d$ due to B. Host.
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