Fourier coefficients of $\times p$-invariant measures

Abstract

We consider densities $D_Σ(A)$, $\overline{D}_Σ(A)$ and $\underline{D}_Σ(A)$ for a subset $A$ of $\mathbb{N}$ with respect to a sequence $Σ$ 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) $μ$ is ergodic and there exist a Følner sequence $Σ$ in $\mathbb{N}$ and a nonzero integer $l$ such that $μ$ is $\times (p^j+l)$-invariant for all $j$ in a subset $A$ of $\mathbb{N}$ with $D_Σ(A)=1$; (2) $μ$ is weakly mixing and there exist a Følner sequence $Σ$ in $\mathbb{N}$ and a nonzero integer $l$ such that $μ$ is $\times (p^j+l)$-invariant for all $j$ in a subset $A$ of $\mathbb{N}$ with $\overline{D}_Σ(A)>0$; (3) $μ$ is strongly mixing and there exists a nonzero integer $l$ such that $μ$ 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 $τ$ defined on positive integers with $\lim_{n\to\infty}τ(n)=\infty$, there exists a multiplicative semigroup $S_τ$ of $\mathbb{Z}^+$ containing $p$ such that $|S_τ\cap[1,n]|\leq (\log_p n)^{τ(n)}$ and the Lebesgue measure is the only non-atomic ergodic $\times p$-invariant measure which is $\times q$-invariant for all $q$ in $S_τ$.