Minimal rates of entropy convergence for rank one systems

If $(X,T)$ is a rank one system and $g$ a positive 
concave funtion on $(0,\infty)$ such 
that $g(x)^2 / x^3$ is integrable, then 
limsup $_{n\to\infty}$ $H(\alpha_0^{n-1})$/$g(log_2 n) =\infty$, 
 for all partitions $\alpha$ of $X$ into two sets with 
$\lim_{n\to\infty} \max\{\mu(A)|A\in\alpha_0^{n-1}\}=0$.


Introduction
In this paper we will continue our investigation of minimal entropy convergence rates that was begun in [2]. Let us review some standard notation: If (X, B, µ) is a probability space, T : X → X a measure-preserving transformation and α a finite partition of X, then the n-th refinement of α under T is denoted by α n The following general result was shown in [2]. This theorem gives us a universal lower bound for entropy convergence rates of measure-preserving sytems. It was shown in [3] that the statement of Theorem 1.1 cannot be improved in the given generality. In order to obtain stronger convergence rates we need to impose additional conditions on (X, T ). The following theorem gives us an example of this type. It was stated in [2] and proved in [3]. The question that we intend to answer in the present paper is whether we can still obtain an improvement of Theorem 1.1 if we reduce the assumptions of Theorem 1.2 from 'rank one mixing' to 'rank one'. We will see that this is indeed the case (see Theorem 5.7), but the convergence rates for arbitrary rank one systems are in general weaker than the rate log 2 n given in Theorem 1.2.

Definition of Rank One Transformations
We consider the probability space ([0, 1), B, µ), where we denote by B the σ-algebra of all Lebesgue measurable subsets of [0, 1) and by µ we denote Lebesgue measure on [0, 1). Definition 2.1. An n-tuple τ = (B 0 , . . . , B n−1 ) of pairwise disjoint sets B i ∈ B is called a tower, if µ(B i ) = µ(B j ) for all i, j ∈ {0, . . . , n − 1}. We will always refer to the sets B i as the levels of τ . The union of all the levels will be denoted by |τ |, i.e.
In case that all the levels in a tower τ are half open intervals, we assign to τ a transformation T τ as follows: x + a k+1 − a k , for x ∈ [a k , b k ) and k ∈ {0, . . . , n − 2} x + a 0 − a n−1 , for x ∈ [a n−1 , b n−1 ) x, for x ∈ [0, 1) |τ |.
This definition implies in particular that T τ (B i ) = B i+1 for all i ∈ {0, . . . , n − 2}, The constructive definition of rank one transformations says that any such transformation can be obtained via 'cutting and stacking' of towers of subintervals of [0, 1). So if {τ n } ∞ n=1 is a sequence of towers τ n = (B 0 (n), . . . , B mn (n)), with ∞ n=1 |τ n | = [0, 1) such that each B i (n) is a subinterval of [0, 1] and τ n+1 is obtained from τ n by cutting τ n into vertical subtowers of equal measure and stacking them with possibly some new levels as 'spacers' added in between, then T := lim n→∞ T τ n is a rank one transformation on [0, 1), where the limit is understood to be a pointwise limit. A more precise definition can be found in [4].
For the rest of our discussion we will assume T to be a rank one transformation with a defining sequence of towers {τ n } ∞ n=1 as explained above. It is clear that each τ n is a Rokhlin tower of T , i.e. τ n = (B 0 (n), T B 0 (n), . . . , T m n −1 B 0 (n)). This allows us to abbreviate our notation by substituting B n for B 0 (n). So we have For the proof of the main lemma in Section 4 it will be useful to work only with towers of length 2 k for some k ∈ N and for this reason we need some additional notation. Definition 2.3. For any n ∈ N we define N n := [log 2 m n ], where [x] := max{n ∈ N | n ≤ x}. Furthermore, for all k ∈ N with k ≤ N n we define σ n,k := (B n,k , T B n,k , . . . , T 2 k −1 B n,k ) and µ n,k := µ µ(B n,k ) .
We notice that |σ n,k | = |σ n,j | for all k, j ≤ N n .
3. Some Properties of 01-Names First we will review some of the definitions and propositions that we already discussed in [2] and then we will discuss some additional properties of 01-names that will be essential for the proof of the main lemma in the next section. (The definitions in this paper are actually slightly modified compared to [2], because we will almost exclusively work with 01-names of length 2 k for some k ∈ N.) Definition 3.1. If α is a finite partition of [0, 1) and F ∈ B, then we define the entropy of α restricted to F as With these definitions it is easy to see that α(E) 2 n −1 In order to understand how the measure of A E n (s) depends on the properties of s, we need to define the period of a 01-name. The proofs of the following simple lemmas were given in [2].
Lemma 3.6. Let α be a finite partition of [0, 1), F ∈ B and c > 0 such that Then H F (α) ≥ µ(F ) log 2 1 c . Definition 3.7. For n, i, j ∈ N with 0 ≤ i ≤ j ≤ n − 1 and s ∈ {0, 1} n we denote a subname of s by s j i := (s i , . . . , s j ). We will use the same notation also for finite subnames of doubly infinite sequences s ∈ {0, 1} Z .
Definition 3.11. It is easy to see that for s ∈ {0, 1} n there exists a uniquē s ∈ {0, 1} Z such thats n−1 0 = s and p(s) = q n (s). We calls the extension of s. Furthermore, we define s + :=s 2n−1 n and s − :=s −1 −n . Remark 3.12. The 01-names s + and s − are the unique elements in {0, 1} n with q 2n ((s, s + )) = q 2n ((s − , s)) = q n (s). Furthermore, if r = s + and t = s − , then it is easy to see that q 2n ((s, r)) > n/2 and q 2n ((t, s)) > n/2. Definition 3.13. For E ∈ B, n, k, K ∈ N with K ≤ k ≤ N n and s, r ∈ {0, 1} 2 k we define ρ k (s, r) := min Remark 3.14. From the definition above follows easily that the maps f k,K are symmetric, i.e. f k, Notice: If we do not assume p k (s) ≤ 2 k−1 then it is possible to have ρ k (s, r) = 0 but ρ k (r, s) = 0. This is for example the case for s := (0001) and r := (0010). Using the observations a), b) and c) it is not difficult to see that (s ∼ r) k :⇔ ρ k (s, r) = 0 defines an equivalence relation on S k . The equivalence classes will be denoted by [s] k . Given this definition it is easy to show that whenever p k (s) < p k (r) and s ∈ S k . We also notice that for all s ∈ S k we have s ∼ s + , s ∼ s − and the maps Proof. Let us define c := It is easy to see that for any i ∈ {0, . . . , Using Remark 3.10 it follows that This completes the proof.
Proof. Simple consequence of Lemma 3.15 and the fact that d(s,r) = d(r,s) (see Remark 3.10).

Proof.
According to Remark 3.14 we haves = λ i (ū) andr = λ j (t) for some i, j ∈ {0, . . . , 2 k − 1}. This implies clearly that d(s,r) = d(ū,t) and therefore we can apply Lemma 3.15 to obtain that The statement of the corollary follows now easily from the definition of f k,K and the proof is complete.

The Main Lemma
The most important tool for the proof of Theorem 5.7 is Lemma 4.3. We will try to give an explanation for the significance of this lemma. The proof of Theorem 1.1 as given in [2] involved estimates of the form where x k stood for the measure of the set of points in the base level of a Rokhlin tower of length 2 n with periods ranging between 2 k−1 and 2 k − 1. The nature of the esitmates that led to the statement of Theorem 1.1 can be summarized as follows: and this is a contradiction. The purpose of Lemma 4.3 in the context of this type of estimate is to allow us to replace the assumption This in turn will make it possible to improve the convergence rates {a n } ∞ n=1 by assuming only that ∞ n=1 a n 2 n 3 < ∞.
Under these assumptions we obtain lim sup n→∞ 1 an n k=1 k x k = ∞, because otherwise we can again find a c > 0 such that and this is a contradiction. The estimate above will actually not appear in exactly this form in the proof of Theorem 5.7, but it illustrates nicely the motivation for Lemma 4.3, because there we obtain a lower estimate on the product of sums of the x k , which is best interpreted as a lower estimate for each individual value x k 2 . This will eventually allow us to obtain estimates that are similar to the assumption ∞ k=1 x k 2 = ∞. Finally, we also wish to mention that the original idea for the proof of Lemma 4.3 is related to Kolmogorov's inequality (see [1]). In fact, Kolmogorov's inequality can be used directly to prove a statement analogous to Theorem 5.7 for periodic approximations of the von Neumann-Kakutani adding machine.
Using these definitions, we obtain (4.10) We will now find an upper estimate for I k . Using the symmetry of f k , Remark 3.14, Corollary 3.17 and the fact that f k (s, r) = 0 for all s, r ∈ W , we obtain and for r ∈ W we have µ k (r) = 1 2 t∈{0,1} 2 k (µ k+1 (t, r) + µ k+1 (r, t)).
Using these equalities and the symmetry of f k it is easy to see that Using this estimate, inequality (4.10), the definition of U and V and the fact that f k ≤ 2 −K , we obtain (µ k+1 (D + (i)) + µ k+1 (D − (i)))(µ k+1 (C + (r)) + µ k+1 (C − (r))) Now we will examine the relation of the measure of the sets D + (i), D − (i), C + (r) and C − (r) to the values of x E n,k (l) and x E n,k+1 (l) for l ∈ {0, . . . , 4k}. If l ≤ k, then it is easy to see that ) and Using Remark 3.12 and the fact that p k+1 ((s, r)) ≥ p k (r) for all s, r ∈ {0, 1} 2 k we conclude that for all ).

Conclusion of the Argument
The following two Lemmas will allow us to use estimates of the type mentioned in the introduction to Section 4.
Proof. If r k = r j for all k, j ∈ {−2 n−2 , . . . , −1} with k = j, then we are done. So let us assume that there are k 0 , j 0 ∈ {−2 n−2 , . . . , −1} such that k 0 < j 0 and r k0 = r j0 . Then we have Since p n (s 2 n −1 It is now easy to see that for all k, j ∈ {m − 2 n−1 + 1, . . . , m − q} with k = j we have r k = r j and this completes the proof, because q < 2 n−2 . Lemma 5.2. For all n, k ∈ N and E ∈ B with 3 ≤ k ≤ N n we have It is obvious that the sets N (i) are pairwise disjoint and since the length of σ n,k is 2 k and −2 k−2 ≤ i ≤ 2 k−1 , it is also clear that the sets M (i) are pairwise disjoint and that ϑ is a Rokhlin tower. We also notice that S ∩ |ϑ| = ∅ and the sets |η(i)| are pairwise disjoint, i.e. β is a partition of [0, 1). Using the well known subadditivity for the entropy of finite partitions, Lemma 3.2 and the fact that H(β) < log 2 (4k), we obtain: Using (5.1), it is easy to see that For any s ∈ {0, 1} 2 k with p k (s) ≤ 2 k−2 it is not difficult to show that Lemma 1.11 in [2]).
Using this observation, Lemma 3.5 and the definition of η(i) we obtain This allows us to apply Lemma 3.6 to conclude that and Hence This completes the proof.
The following Remark will be useful in the proof of Lemma 5.4 and Theorem 5.7. Remark 5.3. Let E ∈ B, n, k ∈ N with k < N n and x ∈ B n,k . Then we have two possibilities: (1) x ∈ B n,k+1 , (2) x ∈ T 2 k B n,k+1 .
If in the first case we have s E k (T 2 k x) = s E k (x) + then Remark 3.12 implies that Using this observation it is not difficult to show that Lemma 5.4 shows how the assumption lim n→∞ max{µ(A) | A ∈ α(E) n−1 0 } = 0 is needed for the proof of Theorem 5.7. In a completely ergodic system any non-trivial partition will have this property (see Corollary 5.8). According to our assumption we can find sequences We consider now two cases: In this case we simply set m j := k j and obtain Here we define m j := [i j /4] + 2. Then 4(m j − 2) ≤ i j < 4(m j − 1) and m j < k j . Therefore, we can use Remark 5.3 and (5.3) to conclude that ij l≥γij x E n j ,m j (l) ≥ ε and with the same estimates as in case 1 we obtain It is clear that lim j→∞ m j = ∞ because of (5.2), (5.4) and the definition of m j in cases 1 and 2. Therefore, we finally conclude that This completes the proof.
The following two results are important tools in order to make use of the main lemma in Section 4. Combining the statements of Lemma 5.5 and 5.6 we see that for large enough n we have This will allow us to use the main lemma in order to obtain estimates which are closely related to the assumption ∞ n=1 x n 2 = ∞ as explained in the introduction to Section 4. We also wish to point out that the proof of Lemma 5.6 is the only place where we use the rank one property of the transformation T .
Proof. It is obvious that for all s, r ∈ {0, 1} 2 K we have Therefore, the definition of f K,K (notice the equality of the two indeces) implies that Using Remark 3.14, it is also not difficult to see that It is not difficult to show that this implies (see Lemma 3.4 in [3]) for all ε > 0 and therefore lim n→∞ J n = 0. Since H n ≤ m n J n /2 N n ≤ 2 J n , it follows that lim n→∞ H n = 0. The definition of I E N n ,K (n) implies trivially that I E N n ,K (n) ≤ H n for all n, K ∈ N with K ≤ N n . So for a given K ∈ N we have lim n→∞ I E N n ,K (n) = 0, and the proof is complete. Theorem 5.7. Let g : (0, ∞) → R be a positive concave function such that Proof. First we notice that it is sufficient to show that because for a given g it is not difficult to find a positive concave function h with Since h satisfies the same conditions as g, statement (5.8) is also true for h and therefore lim sup The concavity of g implies that g(x)/x is monotone decreasing and this shows that lim x→∞ g(x)/x exists. If we had lim x→∞ g(x)/x > 0, then Since this is a contracdiction to our assumption, we conclude that We x E ni,νi(j+1) (l).
(5.15) W.l.o.g. we may assume that lim x→∞ log 2 x/g(x) = 0, because In fact the function g typically converges to ∞ like for example x/ ln x which is much faster than log 2 x (to be more precise, we could use a similar argument as for proving the sufficiency of (5.8)). Hence Let us assume that this lim sup is equal to zero. Then we can find an L ∈ N such that for all i ≥ L and all j ∈ {0, . . . , c i − 1} we have Using the fact that g(x)/x is monotone decreasing and the estimate d i < 3/2 (see above), we obtain y i (j) 2 < 16 g(ν i (j + 1) − 1) 2 (ν i (j + 1) − 1) 2 ≤ 16 g(ν i (j) + 1) 2 (ν i (j) + 1) 2 ≤ 16 g(2 ai+dij ) 2 2 2(a i +d i j) Now we apply (5.14) and the assumption N ni < K i+1 /4 to conclude that It follows that Since this is a contradiction, we have shown (5.16) and the proof is complete.

Some Related Questions
It is an open question whether the statement of Theorem 5.7 is the best possible one, but we believe that this is indeed the case. It should be possible to prove this by using the von Neumann-Kakutani adding machine. More precisely, given a g which violates the assumption of the integrability of g(x) 2  where the refinements are generated by using the adding machine. Another question is whether Theorem 5.7 can be generalized to infinite rank transformations. We believe that this is also possible in the sense that the rate g would depend on the growth rate of the number of towers in the construction of T . For infinite rank transformations the rates g would typically be slower than the rates in Theorem 5.7.
Finally we wish to remark that as a consequence of Theorem 5.7 we see that some of the transformations constructed in Chapter 5 of [3] are not rank one transformations. This is so, because these transformations were designed to demonstrate that the statement of Theorem 1.1 is in general the best possible one, i.e. we generated convergence rates that were slower than the statement of Theorem 5.7 would allow.