Riesz capacities of a set due to Dobi\'nski

We study the Riesz $(a,p)$-capacity of the so called Dobi\'nski set. We characterize the values of the parameters $a$ and $p$ for which the $(a,p)$-Riesz capacity of the Dobi\'nski set is positive. In particular we show that the Dobi\'nski set has positive logarithmic capacity, thus answering a question of Dayan, Fernand\'ez and Gonz\'alez. We approach the problem by considering the dyadic analogues of the Riesz $(a,p)$-capacities which seem to be better adapted to the problem.


Introduction and main results
In a series of two papers [10,11] Dobiński claims that the following identity is true n≥0 (tan 2 n πx) 2 −n = (2 sin πx) 2 , for all real numbers x ∈ [0, 1] which are not dyadic rationals.As it has been already noted in [2] and explained in detail in a recent paper [9] the situation is not quite so simple.In fact, if we consider the same identity with absolute values, n≥0 | tan 2 n πx| 2 −n = (2 sin πx) 2 , so as to avoid issues of defining the powers of negative numbers, in [2] the authors prove that the identity holds if and only if x does not belong to the so called Dobiński set D. To define D let x ∈ [0, 1] be a real number with dyadic expansion x = (0.a 1 a 2 . . . ) 2 and for n ≥ 1 let s n (x) = max{r ∈ N : a n = a n+1 = • • • = a n+r } 1 .Then, 2 n > 0}.So the Dobiński set comprises real numbers which can be approximated "exceedingly well" by dyadic rationals on every scale.Related problems of diophantine approximation by dyadic rationals have been considered in [13].
In a recent work [9] Dayan, Fernandéz and González prove, among other results, that D has Hausdorff dimension 0 and logarithmic Hausdorff dimension 1.Their techniques are primarily based on the mass transference principle of Beresnevich and Velani [6] which allows one to transfer measure theoretic statements for lim sup subsets of R n to statements about Hausdorff measure.The Hausdorff dimension is a precise way to talk about the size of a subset of R n .Another way to measure the size of subsets of R n is by some kind of capacity.
From now on it will be convenient to consider D as a subset of the unit circle T in R 2 via the usual correspondence x → e 2πix .Let 0 < a < 1 and f a positive measurable function on T the a-Riesz potential of f is defined as where dy is the normalized Lebesgue measure on T. Finally let 1 < p < ∞ and a as before.The (a, p)-Riesz capacity of a Borel subset E of T is defined as We shall refer to the capacities R a,p as linear if p = 2 and as non-linear if p = 2.We shall also assume that ap ≤ 1, otherwise singletons have positive capacity.There exists a remarkable relation between Hausdorff dimension, which we will denote by dim, and linear Riesz capacities for a Borel set E ⊆ T established by Frostman [1,Corollary 5.1.14]dim E = sup{1 − ap : R a,p (E) > 0}. ( This fact, together with the standard comparison results between capacities [1, Section 5.5] and the fact that the Dobiński set has vanishing Hausdorff dimension implies that R a,p (D) = 0 when ap < 1.In the same work where the Hausdorff dimension of D was studied the authors ask whether also R1 2 ,2 (D) = 0 or not [9, Section 5].In fact they formulate the question in terms of logarithmic capacity in the complex plane but it is well known that for subsets of T, logarithmic capacity is bounded below and above by the Riesz ( 1 2 , 2)-capacity [7, Corollary 2.6].
We have been able to answer the above question for all Riesz ( 1 p , p)-capacities.Theorem 1.1.Let D be the Dobiński set and p > 1.Then, Somewhat surprisingly the capacity of D exhibits a jump from full to 0 at the critical value p = 2.It should be mentioned that this statement implies, via (1), that the Hausdorff dimension of D is 0 and that the logarithmic Hausdorff dimension is 1 by [1, Corollary 5.1.14].The proof of the above theorem is presented in Section 3 and it applies to a more general class of Dobiński type sets and all (a, p) Riesz capacities (see Theorem 3.2).The proof rests on two ideas.One is the use of a discrete/dyadic version of the Riesz capacity.Discrete type capacities have appeared in potential theory in the past (see for example [5,14]) and their "combinatorial" nature suites very well the dyadic structure of D. In concrete terms one can show using a recursive formula (Lemma 2.2) that D has positive discrete capacity and this, through a comparison theorem for discrete and Riesz capacities, [4, Theorem 1] allow one to deduce that D has positive Riesz ( 1 p , p)-capacity for 1 < p ≤ 2 and zero capacity when p > 2. Finally we prove a "Kolmogorov 0 − 1" type lemma (Lemma 3.1) from which we can deduce that in fact D is of full capacity, i.e.R 1 p ,p (D) = R 1 p ,p (T) when 1 < p ≤ 2. It is worth noticing that the same phenomenon (0 − 1 type law) appears in the study of logarithmic capacity of uniform G δ -sets [12, Theorem 1.2].

Trees, dyadic capacity and the recursion formula
Let T := {0, 1} * the free monoid generated by the language {0, 1} with neutral element e.In this context we shall call T the dyadic tree.The length of a word x is denoted |x|.For two words x, y ∈ T we denote the largest common prefix of x and y by x ∧ y.If x ∧ y = x we write x ≤ y.Finally we use the notation x − = x0, x + = x1.The (Poisson) boundary ∂T of T can be identified with the metric space {0, 1} N equipped with the metric d(x, y) := 2 −|x∧y| .
We will write T := ∂T ∪ T.There exists a natural mapping from ∂T to [0, 1], which is onto and Lipschitz continuous.Moreover, every x ∈ [0, 1] which is not a dyadic rational has a unique pre-image.Dyadic rationals have two pre-images under Λ.
Our next goal is to develop a potential theory on ∂T which parallels the one we have already seen in T. A more detailed exposition of the potential theory on the boundary of the tree can be found in [4,8,3].Here we shall present only the elements that are essential for our problem.
Let ϕ a non negative function defined on T .The potential of ϕ is given by Let π be a positive weight function defined on T .Then for a set E ⊆ ∂T we define its π (discrete) capacity as follows cap π (E) := inf x∈T ϕ(x) p π(x) : ϕ ≥ 0 and Iϕ ≥ 1 on E .
When π(x) = 2 −|x|(1−ap) we shall refer to the capacity cap π =: cap a,p as discrete (a, p) capacity.The relation between the Riesz and discrete capacities can be made explicit.This has been first noted in [5] and generalized further in [4].
Theorem 2.1.[4, Theorem 1] Let p > 1, 0 < a ≤ 1/p.There exists a constant c = c(a, p) > 0 such that for any compact set In fact the restriction that K should be a compact set can be relaxed considerably.By Choquet's capacitability theorem [1,Theorem 2.3.11], Theorem 2.1 holds for all Suslin sets, in particular for all Borel sets.
Discrete capacities satisfy a recursive formula, which is of fundamental importance for our computations.It relates the capacity of a set to the capacities of the parts of its dyadic decomposition.Let x ∈ T and E ⊆ ∂T.Let also E x := {w ∈ ∂T : xw ∈ E} and π x (w) = π(xw).Then we define Informally, cap π (E, x) is the capacity of the portion of E that stays below x "viewed" from the root x.Theorem 2.2.[4, Theorem 30] Let E ⊆ ∂T a Borel set.For every x ∈ T the following equality holds Finally let us introduce a more general class of Dobiński type sets on the boundary of the tree.This is a rather natural generalization of the set D. Suppose that κ n is a sequence of positive integers.Let D(n, κ n ) := {w ∈ ∂T : w n+1 = w n+2 = . . .w n+κn = 0}.We define the Dobiński type set associated to κ n as Notice that if we consider the set Λ(∪ m D m ) where D m is the Dobiński type set corresponding to the sequence κ n = 2 n m −1 , we obtain "one half" of the Dobiński set D. The other half is obtained by considering the same construction, where instead of "strings of 0's" in the definition of D(n, κ n ) we consider strings of 1's.

Proof of the main result
Lemma 3.1.Let p > 1 and a > 0 such that ap ≤ 1 and E ⊆ T a Borel set which is invariant under rotations by angles θ, where θ is a dyadic rational number.Then, either R a,p (E) = R a,p (T) or R a,p (E) = 0.
Proof.Assume that R a,p (E) = 0.By Theorem [1, Theorem 2.3.10]there exists a unique non negative function f E ∈ L p (T) such that and Let θ a dyadic rational and define ρ θ f (x) := f (e 2πiθ x).Then it is clear that 4) and ( 5) and by uniqueness ρ θ f E = f E .Since θ is dense in [0, 1] a calculation with the Fourier coefficients of f E shows that f E = c Lebesgue a.e. on T for some positive constant c.By equation ( 5) and the fact that 0 < R a,p (E) ≤ R a,p (T) we get 0 < c p ≤ R a,p (T).Finally let y 0 ∈ E, such that (4) holds; We now turn to the main theorem.The calculation can be carried out for general (a, p), Theorem 3.2.Let D a Dobiński set associated to a sequence κ n and a > 0, 1 < p < ∞.In the case ap = 1 we have Proof.Let D(n, κ n ) as before.We start with deriving an exact formula for the discrete (a, p) capacity of the set D(n, κ n ) using the recursive formula (equation ( 3)).For a positive parameter r > 0 define the function An elementary computations shows that the following semigroup law is satisfied Next we apply n + κ n times the recursive formula (3) for the set D(n, κ n ).In the following c := cap a,p (∂T ).If we use the symbol for repeated composition of functions we have where we have used the fact that Φ r (2x) = 2Φ Consequently if ap = 1, From which is easily verified that there exists a constant A > 0 such that