Fourier coefficients of $\times p$-invariant measures

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 F\o lner 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 F\o lner 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 $|S_\tau\cap[1,n]|\leq (\log_p n)^{\tau(n)}$ and the Lebesgue measure is the only non-atomic ergodic $\times p$-invariant measure which is $\times q$-invariant for all $q$ in $S_\tau$.


Introduction
There are two motivations for this paper. Both are related to the celebrated ×p, ×q conjecture by H. Furstenberg. The first motivation is Lyons' Theorem and Rudolph-Johnson's Theorem, and the second is a theorem due to E. A. Sataev and later independently discovered by M. Einsiedler and A. Fish.
For an integer p, consider the group homomorphism T p (called the ×p map) on the unit circle T = R/Z given by T p (x) = px mod Z for all x in R/Z.
When p and q are positive integers greater than 1 with log p log q / ∈ Q, H. Furstenberg gave a classification of ×p, ×q-invariant closed subsets in T [Fur67, Thm. IV.1].
Theorem 1.1. [Furstenberg, 1967] A ×p, ×q-invariant closed subset in T is either finite or T.
Motivated by this, H. Furstenberg conjectured a classification of ×p, ×q-invariant measures.
Conjecture. [Furstenberg's ×p, ×q conjecture] For two positive integers p, q ≥ 2 with log p log q / ∈ Q, an ergodic ×p, ×q-invariant measure on T is either finitely supported or the Lebesgue measure. That is, the only nonatomic ×p, ×q-invariant measure on T is the Lebesgue measure. Theorem 1.2. [Lyons' theorem] Suppose p, q are relatively prime. The Lebesgue measure is the only non-atomic ×p, ×q-invariant measure which is T p -exact.
A measure µ is T p -exact means h µ (T p , ξ) > 0 for any nontrivial finite partition ξ of T, where h µ (T p , ξ) stands for the measure entropy of T p with respect to a finite partition ξ. Suppose that log p log q / ∈ Q. Then a ×p, ×q-invariant measure with h µ (T p ) > 0 is the Lebesgue measure.
In this paper by assuming that a non-atomic ×p-invariant measure µ satisfies weaker conditions than T p -exactness or positive entropy, we prove that if µ is invariant under enough many ×q-maps of special forms, then µ is the Lebesgue measure.
Theorem 5.1. The Lebesgue measure is the only non-atomic ×p-invariant measure on T satisfying one of the following: (1) it is ergodic and there exist a nonzero integer l and a Følner sequence Σ = {F n } ∞ n=1 in N such that µ is ×(p j + l)-invariant for all j in some A ⊆ N with D Σ (A) = 1; (2) it is weakly mixing and there exist a nonzero integer l and a Følner sequence (3) it is strongly mixing and there exist a nonzero integer l and an infinite set Moreover, a ×p-invariant measure satisfying (2) or (3) is either a Dirac measure or the Lebesgue measure.
Here D Σ (A) and D Σ (A) are density and upper density of A with respect to Σ respectively. See Section 2 for their definitions. if a Borel probability measure on T is an ergodic ×p-invariant measure for some p in S and is ×q-invariant for every q in S, then it is either finitely supported or Lebesgue measure.
As an application of Theorem 5.1, we prove that there exists a multiplicative semigroup S of positive integers with lim n→∞ log |S ∩ [1, n]| log n = 0 such that Theorem 1.5 still holds (see Theorem 5.3).
The paper is organized as follows.
Firstly we give definitions of density functions of a subset A of nonnegative integers with respect to a Følner sequence. In Section 3, we lay down some basic facts about Fourier coefficients of a measure on the unit circle. In Section 4, we give the characterizations of ergodic, weakly mixing and strongly mixing ×p-invariant measures via their Fourier coefficients. In the last section, we prove the main theorem, Thanks to suggestions of the anonymous referee and Anatole Katok, various changes are made. In particular, they kindly helps me to improve Theorem 5.3.

Preliminaries
Let N stand for the set of nonnegative integers and Z + stand for the set of positive integers. Throughout this article, for two integers a < b, we denote the set {a, · · · , b} by [a, b]. Denote by |F | the cardinality of a set F .
Generalizing these, one can define densities of A with respect to every sequence of finite subsets of N. (2) The density Within this paper, a measure on a compact metrizable X always means a Borel probability measure. A measure µ is called non-atomic if µ{x} = 0 for every x in X.
A topological dynamical system consists of a compact metrizable space X and a continuous map T : X → X.
Within this paper, we only consider that X = T and T = T p is the ×p map on

Some basic facts about Fourier coefficients
Denote the support of µ by Supp(µ). For n in Z, the Fourier coefficientμ(n) of a measure µ on T is given byμ(n) = T z n dµ(z) when taking T = {z ∈ C||z| = 1}. Proof. Obvious.
Proof. Let k be a nonzero integer. By Lemma 3.1, it suffices to show that |μ(k)| < 1 if and only if T |z k − c| 2 dµ(z) > 0 for all c ∈ T.
If |μ(k)| < 1, then for any c ∈ T, we have 4. Fourier coefficients of ergodic, weakly mixing or strongly mixing ×p-invariant measures In this section, we give characterizations of ergodic, weakly mixing and strongly mixing ×p-invariant measures via their Fourier coefficients.
Theorem 4.1. The following are true.
(1) A measure µ on T is an ergodic ×p-invariant measure if and only if for every f in L 2 (X, ν) (note that the identity holds with respect to L 2 -norm). Consequently (1) Suppose µ is an ergodic ×p-invariant measure on T. Denote the ×p map by T p . Consider the measurable dynamical system (T, T p , µ). Using Lemma 4.2, we get for all continuous functions f, g on T. By choosing f = z k and g = z l , we prove the necessity. Now assume that lim for every k in Z. Replacing k by kp, one haŝ g dµ for all f, g ∈ L 2 (T, µ).
In particular, it is true for f = g = 1 A for a Borel subset A with T −1 p A = A. Hence µ(A) = µ(A) 2 . This proves that µ is ergodic.
Note that for all k 1 , k 2 , l 1 , l 2 in Z.
This completes the proof.
for all Borel subsets A and B, where 1 A stands for the characteristic function of A.
Note that linear combinations of characteristic functions are dense in L 2 (T, µ), so lim j→∞ f (T j p x)g(x) dµ(x) = T f dµ T g dµ for all f, g in C(T). In particular, this holds for f = z k and g = z l for all k, l in Z, which means lim j→∞μ (kp j + l) =μ(k)μ(l) for all k, l ∈ Z.
On the other hand, if a measure µ satisfies that lim j→∞μ (kp j + l) =μ(k)μ(l) for all k, l ∈ Z. Let l = 0 and replace k by kp. Then we havê Linear combinations of z k and z l are polynomials on T, which is dense in L 2 (T, µ). Hence lim j→∞ µ(f (T j p )g) = µ(f )µ(g) for all f, g in L 2 (T, µ). In particular, it holds for f = 1 A and g = 1 B for any Borel subsets A, B of T, which completes the proof.
(1) As shown in [Lyo88], a measure µ is T p -exact iff for every l in Z. Hence T p -exactness is much stronger than being strongly mixing.
(2) So far it is unknown how to characterize that h µ (T p ) > 0 via Fourier coefficients of µ.

Rigidity of ×p-invariant measures
With the above preliminaries, we are ready to prove the main theorem.
Theorem 5.1. The Lebesgue measure is the only non-atomic ×p-invariant measure on T satisfying one of the following: (1) it is ergodic and there exist a nonzero integer l and a Følner sequence (2) it is weakly mixing and there exist a nonzero integer l and a Følner sequence (3) it is strongly mixing and there exist a nonzero integer l and an infinite set A ⊆ N such that µ is ×(p j + l)-invariant for all j in A.
Moreover, a ×p-invariant measures satisfying (2) or (3) is either a Dirac measure or the Lebesgue measure. Proof.
[Proof of the first part of Theorem 5.1] (1) Suppose µ is an ergodic ×p-invariant measure and there exist a nonzero integer l and a Følner sequence Σ = {F n } ∞ n=1 such that µ is ×(p j + l)invariant for all j in some A ⊆ N with D Σ (A) = 1.
If µ is not Lebesgue measure, then there exists nonzero k in Z such that 0μ(k) is nonzero.
(2) Suppose µ is a weakly mixing ×p-invariant measure and there exist a nonzero integer l and a Følner sequence Σ = {F n } ∞ n=1 such that µ is ×(p j +l)-invariant for all j in some A ⊆ N with D Σ (A) > 0.
If µ is not Lebesgue measure, then there exists nonzero k in Z such thatμ(k) is nonzero.
(3) Assume that µ is a strongly mixing ×p-invariant measure and there exist a nonzero integer l and an infinite A ⊆ N such that µ is ×(p j + l)-invariant for all j in A.
If µ is not Lebesgue measure, then there exists nonzero k in Z such thatμ(k) is nonzero.
We finish the proof the first part of Theorem 5.1.
Before proceeding to the proof of the second part of Theorem 5.1, we need a lemma.
An atom for a measure µ on a compact metrizable space X is a point x in X such that µ{x} > 0.
Lemma 5.2. Let T : X → X be a continuous map on a compact metrizable space X. If a T -invariant measure µ has an atom x with µ{x} < 1, then µ is not weakly mixing.
Proof. Suppose µ is weakly mixing and has an atom x with λ = µ{x} < 1. Suppose µ is a measure satisfying (2) or (3). By the first part of Theorem 5.1, if µ is not a Lebesgue measure, then µ has an atom. By Lemma 5.2, we obtain that µ is a Dirac measure at some point z in T.
In particular, there exists a multiplicative semigroup S of Z + containing p and satisfying: (1) lim n→∞ log |S ∩ [1, n]| log n = 0; (2) the Lebesgue measure is the only non-atomic ergodic ×p-invariant measure which is ×q-invariant for all q in S.
Proof. Let {l n } ∞ n=1 be a sequence of positive integers such that lim n→∞ l n = ∞ and define f (m) = m n=1 l n for every positive integer m. Define g(m) = min{log p N |τ (N) ≥ 1 + f (m)} for every positive integer m.
Define F n = [p g(n) , p g(n) + l n ] for every positive integer n. Then Σ = {F n } ∞ n=1 is a Følner sequence in N. Denote ∪F n by A.
Let S τ be the multiplicative semigroup generated by p and p j + 1 for all j ∈ A.
Since D Σ (A) = 1, by (1) of Theorem 5.1, a non-atomic ergodic ×p-invariant measure which is ×q-invariant for all q in S f must be the Lebesgue measure.
Every positive integer n locates in [p g(m) , p g(m+1) ) for some nonnegative integer m.
Note that |{k |p k ≤ n}| ≤ log p n. So for each generator, there are at most log p n choices for its powers.
Since n is in [p g(m) , p g(m+1) ), we have g(m) ≤ log p n. Then 1 + f (m) ≤ τ (n) by the definition of g.
Remark 5.4. Furstenberg's conjecture asks for measure rigidity of a non-lacunary semigroup generated by two positive integers p, q and this semigroup has asymptotically (log n) 2 elements in [1, n]. Sataev, Einsiedler and Fish prove measure rigidity of a semigroup containing asymptotically n α elements in [1, n] for some 0 < α < 1. Theorem 5.3 says that for an arbitrary increasing function τ (n) with lim n→∞ τ (n) = ∞, there is a semigroup with asymptotically (log n) τ (n) elements in [1, n] for which measure rigidity still holds. One can choose τ such that the semigroup S τ is sparsely scattered in Z + .