Fourier coefficients of $\times p$-invariant measures

Abstract

We consider densities $D_{\Sigma }(A)$, ${\overline{D}}_{\Sigma }(A)$ and ${\underline{D}}_{\Sigma }(A)$ for a subset $A$ of $\mathbb{N}$ with respect to a sequence $\Sigma$ of finite subsets of $\mathbb{N}$ and study Fourier coefficients of ergodic, weakly mixing and strongly mixing $\times p$-invariant measures on the unit circle $\mathbb{T}$. Combining these, we prove the following measure rigidity results: on $\mathbb{T}$, the Lebesgue measure is the only non-atomic $\times p$-invariant measure satisfying one of the following: (1) $\mu$ is ergodic and there exist a Følner sequence $\Sigma$ in $\mathbb{N}$ and a nonzero integer $l$ such that $\mu$ is $\times (p^j+l)$-invariant for all $j$ in a subset $A$ of $\mathbb{N}$ with $D_{\Sigma }(A)=1$; (2) $\mu$ is weakly mixing and there exist a Følner sequence $\Sigma$ in $\mathbb{N}$ and a nonzero integer $l$ such that $\mu$ is $\times (p^j+l)$-invariant for all $j$ in a subset $A$ of $\mathbb{N}$ with ${\overline{D}}_{\Sigma }(A)>0$; (3) $\mu$ is strongly mixing and there exists a nonzero integer $l$ such that $\mu$ is $\times (p^j+l)$-invariant for infinitely many $j$. Moreover, a $\times p$-invariant measure satisfying (2) or (3) is either a Dirac measure or the Lebesgue measure. As an application we prove that for every increasing function $\tau$ defined on positive integers with ${\lim }_{n\to \infty }\tau (n)=\infty$, there exists a multiplicative semigroup $S_{\tau }$ of ${\mathbb{Z}}^{+}$ containing $p$ such that $|S_{\tau }\cap [1,n]|\le ({\log }_{p}n{)}^{\tau (n)}$ and the Lebesgue measure is the only non-atomic ergodic $\times p$-invariant measure which is $\times q$-invariant for all $q$ in $S_{\tau }$.

arXiv commentThanks to help of the anonymous referee and Anatole Katok, mistakes are corrected and Theorem 5.3 is improved to a better version. The proof of Theorem 4.1 is simplified. Accepted for publication by Journal of Modern Dynamics