p-ADIC FOLIATION AND EQUIDISTRIBUTION

ELON LINDENSTRAUSS

Abstract. We show that if μ is a measure on ℝ/ℤ ergodic under the ×m map, then μ-a.s. {a_{n}x} is equidistributed, for a significantly larger collection of integer sequences a_{n} than was previously known. In particular, we show that μ-a.s. {r^{n}x} is equidistributed as long as m does not divide any power of r (this was previously known only if r and m are relatively prime). The proof uses the p-adic analogue of results from the theory of smooth dynamical systems.

1. Introduction

The starting point of this paper are theorems by D. Rudolph and A. Johnson [15], [6], and a sharpening of Rudolph’s theorem by B. Host [4]. For a more detailed history of this problem see [13]. Some additional references can be found in the bibliography.

Let r, s be two integers. What D. Rudolph has shown for r, m relatively prime, and A. Johnson for r, m multiplicatively independent (i.e. there is no s ∈ ℤ so that both r and m are powers of s) is that any measure μ on 𝕋 = ℝ/ℤ

1. invariant both under the ×m map and the ×r map,

2. ergodic with respect to the ℤ_{+}^{2} action generated by these maps,

3. and so that μ has positive entropy with respect to the ℤ_{+} action

generated by the ×m map, is Lebesgue measure. This gives partial confirmation to a conjecture of H. Furstenberg from 1967, namely that any measure satisfying only 1 and 2 above is either Lebesgue or is supported on a finite number of points.

Host sharpened Rudolph’s (but not Johnson’s) result, and while the theorems of Rudolph and Johnson show that there are no measures satisfying certain conditions other than Lebesgue measure, Host’s theorem is more constructive in the sense that it proves that a large collection of measures satisfies a certain regularity property. In this paper we show how Johnson’s theorem can be strenghened in the same way:

Date: January 5, 2000.

The author acknowledges the support of NSF grant DMS 97-29992.

Theorem 1.1. Suppose that m does not divide any power of r, and let μ be a measure on 𝕋 invariant and ergodic under the ×m map. Then for μ a.e. x ∈ 𝕋, ({r^{n}x})_{n=1}^{∞} is equidistributed in 𝕋 (with respect to the usual Lebegue measure).

The case m relatively prime to r was proved by Host [4].

While the condition on m and r in Theorem 1.1 is more restrictive than being multiplicatively independent, the general case of Johnson’s theorem still follows, since if m and r are multiplicatively independent the semigroup generated by them contains an m′ and an r′ such that m′ does not divide any power of r′. Another point worth mentioning is that since a (×r, ×m) ergodic measure on 𝕋 need not be ×m ergodic, some (rather simple) argument must be given to show that this theorem does indeed imply Johnson’s theorem. This argument, however, is the same as the one showing that Host’s theorem implies Rudolph’s theorem.

In this formulation the roles of r and m are decidedly different. Indeed, r enters only as a parameter that determines the sequence (r^{n})_{n=1}^{∞}. And so, one may ask for what other sequences (a_{n})_{n=1}^{∞} it is true that for every such μ, for μ-a.e. x, ({a_{n}x})_{n=1}^{∞} is equidistributed. A criterion when one might apply Host’s original proof was found by D. Meiri [13]; roughly the requirement is that as m-adic integers a_{n} are evenly distributed (we denote the m-adic integers by ℤ_{m}). A slight variation of Meiri’s condition is the notion we call having m-adic subexponential cells (Definition 3.1). Meiri also showed that many natural sequences (such as r^{n} + s^{n} where r and s are relatively prime to m, or even r^{s^{n}}) satisfy these conditions.

Unfortunately, when r and m are not relatively prime the distribution of r^{n} in ℤ_{m} is not sufficiently regular for Host’s proof to work. To deal with this sequence, we must take into account the common prime factors of r and m. Let r = p_{1}^{a_{1}} … p_{n}^{a_{n}} and m = q_{1}^{b_{1}} … q_{ℓ}^{b_{ℓ}} be the prime decomposition of r and m respectively, and take m_{1} = ∏_{q_{i} |̸r} q_{i}^{b_{i}}, m_{2} = ∏_{q_{i}|r} q_{i}^{b_{i}}. Now, while the sequence (r^{n})_{n=1}^{∞} is not sufficiently evenly distributed in ℤ_{m} = ℤ_{m_{1}} ⊕ ℤ_{m_{2}}, in ℤ_{m_{1}} this sequence is evenly distributed, and in ℤ_{m_{2}} the sequence (r^{n})_{n=1}^{∞} tends very rapidly to zero. More generally, one can consider sequences a_{n} that even though they are not necessarily evenly distributed in ℤ_{m}, have the property that the conditional distribution of a_{n} ∈ ℤ ⊂ ℤ_{m_{1}} ⊕ ℤ_{m_{2}} in ℤ_{m_{1}} given its projection to ℤ_{m_{2}} (more precisely, the distribution of a_{n} mod m_{1}^{k} given a_{n} mod m_{2}^{k}) is sufficiently regular. The precise condition we need, of having (almost) m_{1}-adic subexponential cells rel m_{2}, is given in Definition 3.1. Naturally, this is satisfied in the case of {r^{n}} where r, m, m_{1}, m_{2} are as above, since then a_{n} mod m_{2}^{k} is essentially constant — only a small finite number of the a_{n} are not congruent to 0 mod m_{2}^{k}. Our main result is that as long as m_{1} ≠ 0, such sequences are m-Host:

Theorem 1.2. Let {a_{i}} be a sequence of m_{1}-adic subexponential cells relative to m_{2}. Then for any measure μ invariant and ergodic under ×m (with m = m_{1}m_{2}) of positive entropy, for μ-a.e. x, {a_{n}x} is equidistributed in 𝕋 (with respect to Lebesgue measure).

As was already observed by A. Katok and R. Spatzier in [8], one can consider the two sided extension 𝕋̄_{m} of (𝕋, ×m) as a foliated space, with two sub-foliations corresponding to m_{1} and m_{2} respectively. For example, the leaf of the m_{1}-foliation containing the point (…, a_{−2}, a_{−1}, a_{0}, a_{1}, a_{2}, …) ∈ 𝕋̄_{m} (with each a_{i} ∈ {0, …, m − 1}) is the set of all (…, b_{−2}, b_{−1}, b_{0}, b_{1}, b_{2}, …) such that:

1. a_{i} = b_{i} for all i larger than some N.

2. for every M < N,

∑_{i=M}^{N} m^{−i+N}a_{i} ≡ 0 (mod m_{2}^{N−M+1}).

Note that each leaf is isomorphic in a natural way to ℚ_{m_{1}}.

To be able to apply a modified version of Host’s proof in this case, we need first to define the entropy related to the m_{1}-adic foliation, and then show that if the entropy of μ is positive the same is true for the part of entropy related to this foliation. The first step is proving the analog of a theorem of Ledrappier and Young [10] on the relationship between entropy, Hausdorff dimension and Liapunov exponents for diffeomorphism (i.e. smooth dynamical systems). Our proof is similar in some respects to the proof of a special case of Ledrappier and Young’s result by R. Kenyon and Y. Peres in [9]. Then one uses the specific geometry of the space to give additional constraints on how the entropy is divided according to these two sub foliations.

An alternative approach to extending Host’s theorem can be found in [11] via convolutions. This approach can applied more generally than the method we develop here, but does not give the equidistribution of {a_{n}x} but only a weaker average result (there are also other applications of the convolution results of [11] that are not related to Host’s theorem).

2. p-adic foliation

Let m = m_{1}m_{2}, with (m_{1}, m_{2}) = 1. If x ∈ 𝕋, we will identify x with its expansion to base m. We will use the notation x_{a...b} = x_{a} … x_{b} (considered either as a string of b − a + 1 digits or an element of ℤ/m^{b−a+1}ℤ) and x_{1...b}^{(i)} = x_{1...b} mod m_{i}^{b} for i = 1, 2. We set α_{a...b} to be the σ-algebra generated by the function x ↦ x_{a...b} and α_{a...b}^{(i)} the σ-algebra generated by x ↦ x_{a...b}^{(i)}. Note that unless b = ∞, both α_{a...b} and α_{a...b}^{(i)} are finite algebras of sets. All these definitions can be equally well applied when m = m_{1}m_{2} … m_{d} with the m_{ℓ} pairwise relatively prime; in this case i can have the values 1 … d. If α is a finite algebra of sets, and β is a σ-algebra, we set

μ(α|β)(x) = E(1_{α(x)}(y)|β)(x),

where α(x) denotes the set in α containing x.

Theorem 2.1. Let μ be a m-invariant measure on 𝕋, with m = m_{1}m_{2}, (m_{1}, m_{2}) = 1. Then for a.e. x the following limits exist and are constant a.e.:

h_{1} = lim \frac{− log μ(α_{1...n}^{(1)} | α_{n+1...∞})}{n}  (2.1)
h_{2} = lim \frac{− log μ(α_{1...n}^{(2)} | α_{1...n}^{(1)} ∨ α_{n+1...∞})}{n}.  (2.2)

These limits exist also in L^{1}(μ). Furthermore, if h(μ) is the entropy of μ with respect to the times m map then h(μ) = h_{1} + h_{2}.

Remark: this theorem is in complete analogy to (one part of) a theorem of Ledrappier and Young on the relations between entropy, Lyapunov exponents and Hausdorff dimension for diffeomorphism. There one uses the subfoliation of the stable or unstable manifolds to divide the entropy into parts corresponding to every positive (or every negative) Lyapunov exponent; here we divide the entropy into parts corresponding to the p-adic foliations modulo m_{1} and m_{2}.

The following lemma is completely standard:

Lemma 2.2. Let P be a finite partition of a measure space (X, B, μ). Then for any set E ⊂ X,

− ∫_{E} log μ(P(x)) ≤ −μ(E) log μ(E) + μ(E) log |P|.

Proof. Indeed, set P′ = P ∨ {E, E^{C}}. The lemma follows from the following calculation

− ∫_{E} log μ(P(x)) ≤ − ∫_{E} log μ(P′(x))

= − ∑_{P ∈P} μ(P ∩ E) log μ(P ∩ E)

= −μ(E) log μ(E) − μ(E) ∑_{P ∈P} μ(P |E) log μ(P |E)

≤ −μ(E) log μ(E) + μ(E) log |P|.

where μ(P |E) denotes

μ(P |E) = \frac{μ(P ∩ E)}{μ(E)}.

Proof of Theorem 2.1. By a variant of the Shannon-McMillen-Brieman theorem,

\frac{− log μ(α_{1...n} | α_{n+1...∞})}{n} ⟶ h(μ)

both pointwise and in L^{1}. Furthermore,

μ(α_{1...n} | α_{n+1...∞}) = μ(α_{1...n}^{(1)} | α_{n+1...∞})μ(α_{1...n}^{(2)} | α_{1...n}^{(1)} ∨ α_{n+1...∞}),

and so it suffice to prove that the limit in (2.2) exists.

Consider now the solenoid 𝕋̄_{m}, i.e. the two sided extension of (𝕋, ×m). Let T denote the extension of the ×m map to 𝕋̄_{m}. The measure μ can be extended to a T invariant measure μ̄ on 𝕋̄_{m}. We can extend our notation x̄_{a...b} also to x̄ ∈ 𝕋̄_{m} (note that a and b may be negative), and again we set

x̄_{a...b}^{(i)} = x̄_{a...b} mod m_{i}^{b−a+1},

and let ᾱ_{a...b} and ᾱ_{a...b}^{(i)} be the corresponding σ-algebras. Notice that for a < a′ < b,

x̄_{a′...b}^{(i)} = x̄_{a...b}^{(i)} mod m_{i}^{b−a′+1}.

Thus ᾱ_{a...b}^{(i)} refines ᾱ_{a′...b}^{(i)}, and we define

ᾱ_{−∞...b}^{(i)} = ⋁_{a=0}^{∞} ᾱ_{−a...b}^{(i)}.

Set

ρ(x̄) = − log μ̄(ᾱ_{0}^{(2)} | ᾱ_{−∞...0}^{(1)} ∨ ᾱ_{1...∞})(x̄).

Clearly ρ(x̄) is non-negative and in L^{1}; indeed, ∫ ρ(x̄)dμ̄(x̄) ≤ log m_{2}. We note that by the Chinese remainder theorem

ᾱ_{−∞...n}^{(1)} ∨ ᾱ_{n+1...∞} ∨ ᾱ_{k+1...n}^{(2)} = ᾱ_{−∞...k}^{(1)} ∨ ᾱ_{k+1...∞},

and so we see that

− log μ̄(ᾱ_{1...n}^{(2)} | ᾱ_{−∞...n}^{(1)} ∨ ᾱ_{n+1...∞})(x̄) = − log μ̄(ᾱ_{n}^{(2)} | ᾱ_{−∞...n}^{(1)} ∨ ᾱ_{n+1...∞})(x̄)

− log μ̄(ᾱ_{n−1}^{(2)} | ᾱ_{−∞...n−1}^{(1)} ∨ ᾱ_{n...∞})(x̄)

⋮

− log μ̄(ᾱ_{1}^{(2)} | ᾱ_{−∞...1}^{(1)} ∨ ᾱ_{2...∞})(x̄)

= ∑_{i=1}^{n} ρ(T^{i}x̄).

(2.3)

By the ergodic theorem the limit

L(x̄) = lim \frac{− log μ̄(ᾱ_{1...n}^{(2)} | ᾱ_{−∞...n}^{(1)} ∨ ᾱ_{n+1...∞})(x̄)}{n}

exists both pointwise and in L^{1}; furthermore, L(x̄) = L is a constant a.e. Thus the theorem will follow once we show that a.e.

lim_{n→∞} \frac{1}{n} |log \frac{μ(ᾱ_{1...n}^{(2)} | ᾱ_{1...n}^{(1)} ∨ ᾱ_{n+1...∞})(x̄)}{μ(ᾱ_{1...n}^{(2)} | ᾱ_{−∞...n}^{(1)} ∨ ᾱ_{n+1...∞})(x̄)}|

Let

Z_{n} = \frac{μ(ᾱ_{1...n}^{(2)} | ᾱ_{1...n}^{(1)} ∨ ᾱ_{n+1...∞})(x)}{μ(ᾱ_{1...n}^{(2)} | ᾱ_{−∞...n}^{(1)} ∨ ᾱ_{n+1...∞})(x)}

Then

E(Z_{n}) = E(E(Z_{n} | ᾱ_{−∞...n}^{(1)} ∨ ᾱ_{n+1...∞}))

= E( ∑_{a∈ℤ/m_{2}^{n}} μ({x̄_{1...n}^{(2)} = a} | ᾱ_{1...n}^{(1)} ∨ ᾱ_{n+1...∞}) | ᾱ_{−∞...n}^{(1)} ∨ ᾱ_{n+1...∞})

= 1.

Therefore, for every ϵ > 0, the series ∑_{k=1}^{∞} μ(Z_{k} ≥ e^{ϵk}) converges and the Borel-Cantelli lemma implies that \overline{lim}_{k→∞} \frac{1}{k} log Z_{k} ≤ ϵ a.s., and so since ϵ is arbitrary,

\underline{lim}_{n→∞} \frac{− log μ(ᾱ_{1...n}^{(2)} | ᾱ_{1...n}^{(1)} ∨ ᾱ_{n+1...∞})(x̄)}{n} ≥

lim_{n→∞} \frac{− log μ(ᾱ_{1...n}^{(2)} | ᾱ_{−∞...n}^{(1)} ∨ ᾱ_{n+1...∞})(x̄)}{n} = L.

To see the opposite inequality, consider

ρ_{ℓ}(x̄) = − log μ̄(ᾱ_{0}^{(2)} | ᾱ_{−ℓ...0}^{(1)} ∨ ᾱ_{1...∞}).

Since for any a the family of random variables μ({x̄_{0}^{(2)} = a} | ᾱ_{−ℓ...0}^{(1)} ∨ ᾱ_{1...∞}) for ℓ = 0, 1, … is a martingale, by the martingale convergence theorem, ρ_{ℓ}(x̄) → ρ(x̄) as ℓ → ∞ for a.e. x̄. To see that ρ_{ℓ}(x̄) → ρ(x̄) also in L^{1}, it suffice to show that the ρ_{ℓ} are uniformly integrable. Let Q be of small measure (say μ̄(Q) = δ). We cover Q by two sets:

Q′ = {x: E(1_{Q} | ᾱ_{−ℓ...0}^{(1)} ∨ ᾱ_{1...∞})(x) > δ^{1/2}} ,

and Q′′ = Q\Q′. Note that μ(Q′) is at most δ^{1/2}. We need to show that if δ is small then ∫_{Q} ρ_{ℓ}(x̄)dμ̄(x̄) is small, uniformly in ℓ. This follows from

∫_{Q′} ρ_{ℓ}(x̄)dμ̄(x̄) = − ∫_{Q′}[∑_{a} μ̄(ᾱ_{0}^{(2)} | ᾱ_{−ℓ...0}^{(1)} ∨ ᾱ_{1...∞})(x) ×

× log μ̄(ᾱ_{0}^{(2)} | ᾱ_{−ℓ...0}^{(1)} ∨ ᾱ_{1...∞})(x)]dμ̄(x)
≤ μ̄(Q′) log m_{2} ≤ δ^{1/2} log m_{2},

and, using Lemma 2.2

∫_{Q′′} ρ_{ℓ}(x) ≤ −δ^{1/2} log δ^{1/2} + δ^{1/2} log m_{2}.

Thus both of these integrals tend to 0 as δ → 0, uniformly in ℓ.

Fix ϵ > 0 and using the above discussion we find an ℓ such that ∥ρ_{ℓ} − ρ∥_{1} < ϵ. We now compare − log μ(ᾱ_{1...n}^{(2)} | ᾱ_{−∞...n}^{(1)} ∨ ᾱ_{n+1...∞})(x) with ∑_{i=1}^{n} ρ_{ℓ}(T^{i}x̄).

First note that by the ergodic theorem,

\frac{1}{n} ∑_{i=1}^{n} ρ_{ℓ}(T^{i}x̄) → ∫ ρ_{ℓ}(ȳ) dμ̄(ȳ)

which is within ϵ from L. Set

R_{n} = \frac{∏_{i=1}^{n} exp(ρ_{ℓ}(T^{i}x̄))}{μ̄(ᾱ_{1...n}^{(2)} | ᾱ_{−ℓ+1...n}^{(1)} ∨ ᾱ_{n+1...∞})(x̄)}.

As before, it is easy to see that E(R_{n}) = 1, and so \overline{lim}_{n→∞} \frac{1}{n} log R_{n} ≤ 0 a.s. Combined with ∥ρ_{ℓ} − ρ∥_{1} < ϵ we get

\overline{lim} \frac{− log μ̄(ᾱ_{1...n}^{(2)} | ᾱ_{−ℓ+1...n}^{(1)} ∨ ᾱ_{n+1...∞})(x̄)}{n} ≤ L + ϵ.

To conclude the proof, we apply the same trick once again with

W_{n} = \frac{m_{2}^{−ℓ}μ̄(ᾱ_{1...n}^{(2)} | ᾱ_{−ℓ+1...n}^{(1)} ∨ ᾱ_{n+1...∞})(x̄)}{μ̄(ᾱ_{−ℓ+1...n}^{(2)} | ᾱ_{−ℓ+1...n}^{(1)} ∨ ᾱ_{n+1...∞})(x̄)}.

Again we see that E(W_{n}) = 1 for which we deduce that a.s. for all n large enough,

− \frac{1}{n} log μ̄(ᾱ_{1...n}^{(2)} | ᾱ_{−ℓ+1...n}^{(1)} ∨ ᾱ_{n+1...∞})(x̄) + \frac{ℓ}{n} log m_{2} ≥

≥ −\frac{1}{n} log μ̄(ᾱ_{−ℓ+1...n}^{(2)} | ᾱ_{1...n}^{(1)} ∨ ᾱ_{n+1...∞})(x̄) − ϵ

Thus for a.e. x̄

L ≤ \underline{lim}_{n→∞} \frac{− log μ̄(ᾱ_{−ℓ+1...n}^{(2)} | ᾱ_{−ℓ+1...n}^{(1)} ∨ ᾱ_{n+1...∞})(x̄)}{n}

≤ \overline{lim}_{n→∞} \frac{− log μ̄(ᾱ_{1...n}^{(2)} | ᾱ_{−ℓ+1...n}^{(1)} ∨ ᾱ_{n+1...∞})(x̄)}{n} + 2ϵ

≤ L + 3ϵ.

Corollary 2.3. Let μ be a m-invariant measure on 𝕋, with m = m_{1}m_{2} … m_{d} and the m_{i} pairwise relatively prime. Then for almost every x and 1 ≤ i ≤ d the following limit exists and is constant a.e.:

h_{i} = lim \frac{− log μ(α_{1...n}^{(i)} | ⋁_{j=1}^{i−1} α_{1...n}^{(j)} ∨ α_{n+1...∞})}{n}.

Furthermore, h(μ) = ∑_{i=1}^{d} h_{i}.

Proof. Let

I_{n}^{(i)}(x) = − log μ (α_{1...n}^{(i)} | ⋁_{j=1}^{i−1} α_{1...n}^{(j)} ∨ α_{n+1...∞}) .

Note that for k ≤ d

∑_{i=1}^{k} I_{n}^{(i)}(x) = − log μ (⋁_{i=1}^{k} α_{1...n}^{(i)} | α_{n+1...∞}) .  (2.4)

In particular, for d = n we get that

\frac{1}{n} ∑_{i=1}^{n} I_{n}^{(i)}(x) = −\frac{1}{n} log μ(α_{1...n} | α_{n+1...∞}) ⟶ h(μ),

so if we prove that the limits defining the h_{i} exist and are constant a.e. it follows that ∑_{i=1}^{d} h_{i} = h(μ).

Moreover, from Theorem 2.1 applied to m̄_{1} = m_{1} … m_{k} and m̄_{2} = m_{k+1} … m_{d} it follows that the limit

lim \frac{1}{n} log μ (⋁_{i=1}^{k} α_{1...n}^{(i)} | α_{n+1...∞})

exists and is constant a.e. Using equation (2.4) it follows that \frac{1}{n}I_{n}^{(i)}(x), as the difference between two functions converging a.e. to constants, converges a.e. to a constant which is by definition h_{i}.

Theorem 2.4. Let h_{i} be as in Corollary 2.3. Then for any k ≤ d,

∑_{i=k}^{d} h_{i} ≤ \frac{∑_{i=k}^{d} log m_{i}}{log m} h(μ)

In particular, since ∑_{i=1}^{d} h_{i} = h(μ), if h(μ) > 0 so is h_{1}.

Proof. Set m̄ = ∏_{i=k}^{d} m_{i}. We first note that for any b, if ℓ = ⌈\frac{log m̄}{log m}b⌉,

then given x_{b+1...∞} and x_{1...b}^{(i)} for all i < k, the value of x_{1...ℓ} determines x_{1...b}^{(j)} for all j ≥ k. Indeed, those y ∈ 𝕋 for which y_{1...b}^{(i)} = x_{1...b}^{(i)} for i < k and y_{b+1...∞} = x_{b+1...∞} are in a coset of the group

\{\frac{k}{m̄^{b}} : 0 ≤ k ≤ m̄^{b} − 1\}.

Since the distance between every two elements of this group is at least

\frac{1}{m̄^{b}} ≥ \frac{1}{m^{ℓ}} ,

the first ℓ digits of y to base m uniquely determine it.

Now, letting b → ∞, we conclude that

∑_{j=k}^{d} h_{j} ⟵\frac{−1}{b} ∫_{𝕋} log μ (⋁_{j=k}^{d} α_{1...b}^{(j)} | ⋁_{i=1}^{k−1} α_{1...b}^{(i)} ∨ α_{b+1...∞}) (x) dμ(x)

≤\frac{−1}{b} ∫_{𝕋} log μ (α_{1...ℓ} | ⋁_{i=1}^{k−1} α_{1...b}^{(i)} ∨ α_{b+1...∞}) (x) dμ(x)
≤\frac{−1}{b} ∫_{𝕋} log μ(α_{1...ℓ})dμ(x) ⟶ h(μ)\frac{log m̄}{log m},

and so ∑_{j=k}^{d} h_{j} ≤ \frac{log m̄}{log m}h(μ).

3. equidistribution

Definition 3.1. Let {a_{i}} be a sequence of integers, m_{1} and m_{2} relatively prime. Set

A(M, n, r) = {a_{i}: 1 ≤ i ≤ M and a_{i} ≡ r mod m_{1}^{n}}
A(M, n, ∗; s) = {a_{i}: 1 ≤ i ≤ M and a_{i} ≡ s mod m_{2}^{n}}
A(M, n, r; s) = {a_{i}: 1 ≤ i ≤ M, a_{i} ≡ r mod m_{1}^{n} and a_{i} ≡ s mod m_{2}^{n}}.

The sequence a_{i} will be said to have m_{1}-adic (almost) subexponential cells if for every ϵ > 0, for some sequence M_{n} such that M_{n+1}/M_{n} = O(1), for every n there is an Â_{n} such that:

1. |Â_{n}| < ϵ|M_{n}|,

2. |A(M_{n}, n, r) \ Â_{n}|/|M_{n}| = O(m_{1}^{−(1−ϵ)n}) uniformly in r, n.

(We will omit the word almost from now on.)

The sequence a_{i} will be said to have m_{1}-adic subexponential cells rel m_{2} if for every ϵ there are M_{n} and Â_{n} as above such that 2’. |A(M_{n}, n, r; s) \ Â_{n}|/|A(M_{n}, n, ∗; s)| = O(m_{1}^{−(1−ϵ)n}) uniformly in

r, s, n.

Examples:

1. Suppose r is relatively prime to m_{1}m_{2}. Then a_{i} = r^{i} has m_{1}- adic subexponential cells, m_{1}-adic subexponential cells rel m_{2}, as well as m_{1}m_{2}-adic subexponential cells. In this case we can take M_{n} = (m_{1}m_{2})^{n}. Note that in general, each of these conditions implies the previous one.

2. Suppose r is relatively prime to m_{1} but is divisible by every prime factor of m_{2}. Then a_{i} = r^{i} has m_{1}-adic subexponential cells, and since practically all a_{i} 1 ≤ i ≤ M_{n} are divisible by m_{2}^{n} this implies that a_{i} has m_{1}-adic subexponential cells rel m_{2}. However, a_{i} does not have m_{1}m_{2}-adic subexponential cells. This has been the motivating example for this work.

3. More generally, at least when m_{1} is a power of some prime, whenever a_{i} = ∑_{i=1}^{N} c_{k}r_{k}^{i} with c_{k} ∈ ℤ, |r_{1}| < |r_{2}| < … with at least one r_{k} relatively prime to m_{1} (and not equal to ±1) then a_{i} has m_{1}-adic subexponential cells rel m_{2}. The restriction that m_{1} is a power of some prime is not really a restriction from our point of view since we can still apply Theorem 3.2 on this sequence.

4. A discussion of recursion sequences, i.e. sequences such that

∑_{i=0}^{t} c_{i}a_{n−i} = 0 for every n (c_{1}, …, c_{t} ∈ ℤ) can be found in [11]. These examples all follow easily using the techniques of p-adic interpolation that are used to prove Theorem 3.2 of [13].

Theorem 3.2. Let {a_{i}} be a sequence of m_{1}-adic subexponential cells relative to m_{2}. Then for any measure μ invariant and ergodic under ×m (with m = m_{1}m_{2}) of positive entropy, for μ-a.e. x, {a_{n}x} is equidistributed in 𝕋 (with respect to Lebesgue measure).

Proof. Let v ∈ ℤ \ {0}, 0 < ϵ < h(μ)/10, and let M_{n} and Â_{n} be as in Definition 3.1, M_{n−1} ≤ M ≤ M_{n}. Set (for r = 0, …,m_{1}^{n} − 1 or ‘∗’)

Ã(M, n, r; s) = A(M, n, r; s) \ Â_{n}.

Define

G_{M}(x) = ∑_{i=1}^{M} e_{v}(a_{i}x) (with e_{v}(t) = e(t) = exp(2πivt))

Since v will be fixed throughout we suppress it in all notations. Set

G_{M,n,s}(x) = ∑_{a∈Ã(M,n,∗;s)} e(ax);

thus

|G_{M}(x) − ∑_{s=0}^{m_{2}^{n}−1} G_{M,n,s}(x)| < CϵM,

where C = max(M_{ℓ}/M_{ℓ−1}).

Step 1: for any s, t ∈ ℤ/m_{2}^{n}

|G_{M,n,s}(x)| = |G_{M,n,s}(x + t/m_{2}^{n})|.

Thus we can write |G_{M,n,s}(x)| = |G_{M,n,s}|(x_{1...n}^{(1)}; x_{n+1...∞}).

This follows from the fact that for every a, b ∈ Ã(M, n, ∗; s) we have that m_{2}^{n}|a − b, and so e((a − b)(x + t/m_{2}^{n})) = e((a − b)x). Thus

|G_{M,n,s}(x)|^{2} = ∑_{a,b∈Ã(M,n,∗;s)} e((a − b)x) = |G_{M,n,s}(x + t/m_{2}^{n})|.

Step 2:

\begin{aligned}∑_{r=0}^{m_{1}^{n}−1} |G_{M,n,s}(x + r/m_{1}^{n})|^{2} & = ∑_{r=0}^{m_{1}^{n}−1} |G_{M,n,s}|(r; x_{n+1...∞})^{2} \\ & = O_{ϵ}(|A(M_{n}, n, ∗; s)|^{2}m_{1}^{ϵn}). \end{aligned}

Indeed,

∑_{r=0}^{m_{1}^{n}−1} |G_{M,n,s}(x + r/m_{1}^{n})|^{2} = ∑_{r=0}^{m_{1}^{n}−1} ∑_{a,b∈Ã(M,n,∗;s)} e((a − b)(x + r/m_{1}^{n}))

= m_{1}^{n} ∑_{r=0}^{m_{1}^{n}−1} |Ã(M_{n}, n, r; s)|^{2}

= O_{ϵ}(|A(M_{n}, n, ∗; s)|^{2}m_{1}^{ϵn}),

where the last inequality is a consequence of 2’ in Definition 3.1. Step 3: for every x_{n+1...∞}, there are at most O_{ϵ}(m_{1}^{ϵn}) x_{1...n}^{(1)} so that

∑_{s=0}^{m_{2}^{n}−1} |G_{M,n,s}|(x_{1...n}^{(1)}; x_{n+1...∞}) > ϵM.

Indeed, denote the set of these x_{1...n}^{(1)} by V_{M,n}(x_{n+1...∞}). Then (in all summations bellow s runs from 0 to m_{2}^{n} − 1, x_{1...n}^{(1)} on all possible values)

|V_{M,n}(x_{n+1...∞})| ≤ (ϵM)^{−2} ∑_{x_{1...n}^{(1)}} (∑_{s} |G_{M,n,s}|(x_{1...n}^{(1)}; x_{n+1...∞}))^{2}

≤ (ϵM)^{−2} ∑_{x_{1...n}^{(1)}} (∑_{s} |Ã(M_{n}, n, ∗; s)|) ×

× (∑_{s} \frac{|G_{M,n,s}|(x_{1...n}^{(1)}; x_{n+1...∞})^{2}}{|Ã(M_{n}, n, ∗; s)|} )

≤ (ϵM)^{−2}M_{n} ∑_{s} \frac{1}{|Ã(M_{n}, n, ∗; s)|} ×

× ∑_{x_{1...n}^{(1)}} |G_{M,n,s}|(x_{1...n}^{(1)}; x_{n+1...∞})^{2}

= O_{ϵ} (M^{−2}M_{n} ∑_{s} |Ã(M_{n}, n, ∗; s)|m_{1}^{ϵn})

= O_{ϵ}(m_{1}^{ϵn}M_{n}^{2}/M^{2}) = O_{ϵ}(m_{1}^{ϵn}).

Note that all estimates above are uniformly in n, x. Step 4: for a.e. x, if M is large enough,

\frac{1}{M} G_{M}(x) < 10(1 + C)ϵ.  (3.1)

Take I_{ϵ} = {⌊(1 + ϵ)^{τ}⌋} and I_{ϵ,n} = I_{ϵ} ∩ [M_{n−1}, M_{n}]. Set

V_{n}(x_{n+1...∞}) = ∪_{M∈I_{ϵ,n}}V_{M,n}(x_{n+1...∞}).

Note first that in order to prove (3.1) it is clearly enough to show that for all large enough M ∈ I_{ϵ}

\frac{1}{M} G_{M}(x) < 5(1 + C)ϵ.

Furthermore, |V_{n}(x_{n+1...∞})| = O_{ϵ}(m_{1}^{ϵn}).

Let

B_{n} = {y: μ(α_{1...n}^{(1)} | α_{n+1...∞})(y) < 2^{−(h−ϵ)n}} .

By Theorems 2.1 and 2.4, a.e. x is in B_{n} for all n large enough. But now clearly

μ (x ∈ B_{n} and x_{1...n}^{(1)} ∈ V_{n}(x_{n+1...∞})) = O_{ϵ}(m_{1}^{2ϵn}2^{−hn}).

Since for small ϵ these probabilities are summable, by Borel-Cantelli, for all n large enough, either x ∉ B_{n} or x_{1...n}^{(1)} ∉ V_{n}(x_{n+1...∞}). The former is impossible by the theorems cited above, the latter implies that for all M ∈ I_{ϵ,n}

\frac{1}{M} G_{M}(x) ≤ Cϵ + \frac{1}{M} ∑_{s} |G_{M,n,s}(x)| ≤ (1 + C)ϵ.

From Step 4 the theorem follows by using Weyl’s criterion for equidistribution.

Acknowledgments: The author would like to thank H. Furstenberg, D. Rudolph, B. Weiss, L-S. Young and especially D.Meiri and Y. Peres for many discussions on this and related questions.

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School of Mathematics, Institute for Advanced Study, Olden lane, Princeton NJ 08540 and Department of Mathematics, Stanford University, Stanford, CA 94305

E-mail address: elon@ias.edu