Zoltán Buczolich

AnL 1 counting problem in ergodic theory

We give a negative solution to the following counting problem for measure preserving transformations. Forf∈L+1(μ), is it true that supn (Nn(f)(x)/n) <∞, μ a.e., where Nn(f)(x)=≠{k:f(Tkx)/k>1/n}? One of the consequences is the nonvalidity of J. Bourgain’s Return Time Theorem for pairs of (L1,L1) functions.

Variations of additive functions

We study the relationship between derivates and variational measures of additive functions defined on families of figures or bounded sets of finite perimeter. Our results, valid in all dimensions, include a generalization of Wards theorem, a necessary and sufficient condition for derivability, and full descriptive definitions of certain conditionally convergent integrals.

Topological Hausdorff dimension and level sets of generic continuous functions on fractals

In an earlier paper we introduced a new concept of dimension for metric spaces, the so called topological Hausdorff dimension. For a compact metric space K let dimH K and dimtH K denote its Hausdorff and topological Hausdorff dimension, respectively. We proved that this new dimension describes the Hausdorff dimension of the level sets of the generic continuous function on K, namely sup{dimHf-1(y):y∈R}=dimtHK-1 for the generic f ∈ C(K), provided that K is not totally disconnected, otherwise every non-empty level set is a singleton. We also proved that if K is not totally disconnected and sufficiently homogeneous then dimH f−1(y) = dimtH K − 1 for the generic f ∈ C(K) and the generic y ∈ f(K). The most important goal of this paper is to make these theorems more precise. As for the first result, we prove that the supremum is actually attained on the left hand side of the first equation above, and also show that there may only be a unique level set of maximal Hausdorff dimension. As for the second result, we characterize those compact metric spaces for which for the generic f ∈ C(K) and the generic y ∈ f(K) we have dimH f−1(y) = dimtH K − 1. We also generalize a result of B. Kirchheim by showing that if K is self-similar then for the generic f ∈ C(K) for every y∈intf(K) we have dimH f−1(y) = dimtH K − 1. Finally, we prove that the graph of the generic f ∈ C(K) has the same Hausdorff and topological Hausdorff dimension as K.

Typical Borel measures on [0, 1]d satisfy a multifractal formalism

In this paper, we prove that in the Baire category sense, measures supported by the unit cube of typically satisfy a multifractal formalism. To achieve this, we compute explicitly the multifractal spectrum of such typical measures ?. This spectrum appears to be linear with slope 1, starting from 0 at exponent 0, ending at dimension d at exponent d, and it indeed coincides with the Legendre transform of the Lq-spectrum associated with typical measures ?.

Solution to the gradient problem of C. E. Weil

In this paper we give a complete answer to the famous gradient problem of C. E. Weil. On an open set G I R2 we construct a differentiable function f: G ? R for which there exists an open set O1 I R2 such that Nf(p) I O1 for a p I G but Nf(q) I O1 for almost every q I G. This shows that the Denjoy-Clarkson property does not hold in higher dimensions.

On series of translates of positive functions II

Abstract In this paper we continue our investigation of series of the form ∑ λ ∈ Λ ƒ(x + λ) . Given a sequence of natural numbers n1 where 0 α = 1 q , where q > 1 is an integer, there is a zero-one law showing that for every measurable the above sum either converges almost everywhere or diverges almost everywhere. However, for any other value of α ∈ (0, 1) there is no such zero-one law.

Henstock integrable functions are Lebesgue integrable on a portion

If a real function / defined on an interval / C Rm is Henstock integrable, then one can always find a nondegenerate subinterval J C / on which / is Lebesgue integrable.

Monotone and convex restrictions of continuous functions

Abstract Suppose that f belongs to a suitably defined complete metric space C α of Holder α-functions defined on [ 0 , 1 ] . We are interested in whether one can find large (in the sense of Hausdorff, or lower/upper Minkowski dimension) sets A ⊂ [ 0 , 1 ] such that f | A is monotone, or convex/concave. Some of our results are about generic functions in C α like the following one: we prove that for a generic f ∈ C 1 α [ 0 , 1 ] , 0 α 2 for any A ⊂ [ 0 , 1 ] such that f | A is convex, or concave we have dim H A ≤ dim _ M A ≤ max ⁡ { 0 , α − 1 } . On the other hand we also have some results about all functions belonging to a certain space. For example the previous result is complemented by the following one: for 1 α ≤ 2 for any f ∈ C α [ 0 , 1 ] there is always a set A ⊂ [ 0 , 1 ] such that dim H A = α − 1 and f | A is convex, or concave on A.

The (L1,L1) bilinear Hardy-Littlewood function and Furstenberg averages

Let

Continuous functions with everywhere infinite variation with respect to sequences

(X,\mathcal{B}, \mu, T)