Self-affine sets with positive Lebesgue measure

Using techniques introduced by C. Güntürk, we prove that the attractors of a family of overlapping self-affine iterated function systems contain a neighbourhood of zero for all parameters in a certain range. This corresponds to giving conditions under which a single sequence may serve as a ‘simultaneous β$\beta$-expansion’ of different numbers in different bases.     

Some properties of Baire sets and sets of positive measure

In un lavoro classico Steinhaus ha dimostrato undici teoremi interessanti nei quali si considerano insiemi distanziali di sottoinsiemi della linea reale,aventi misura di Lebesgue positiva. Majumder ha dimostrato che sono validi teoremi analoghi a quelli di Steinhaus per insiemi del tipoR(E)={x/y: x,y∈E} doveE∪R; 0?E. In questo lavoro si considerano funzioni generali e si investiga sui quali degli undici teoremi di Steinhaus possono essere generalizzati. Nell'articolo, inoltre, usando una funzione generalef, vengono dati risultati analoghi a quelli di Steinhaus per insiemi di Baire.     

Self-similar sets of zero Hausdorff measure and positive packing measure

We prove that there exist self-similar sets of zero Hausdorff measure, but positive and finite packing measure, in their dimension; for instance, for almost everyu ∈ [3, 6], the set of all sums ∑ ₀ ⁸ a _n 4⁻ⁿ a _n 4⁻ⁿ with digits witha _n ∈ {0, 1,u} has this property. Perhaps surprisingly, this behavior is typical in various families of self-similar sets, e.g., for projections of certain planar self-similar sets to lines. We establish the Hausdorff measure result using special properties of self-similar sets, but the result on packing measure is obtained from a general complement to Marstrand’s projection theorem, that relates the Hausdorff measure of an arbitrary Borel set to the packing measure of its projections. Research of Y. Peres was partially supported by NSF grant #DMS-9803597. Research of K. Simon was supported in part by the OTKA foundation grant F019099. Research of B. Solomyak was supported in part by NSF grant #DMS 9800786, the Fulbright Foundation, and the Institute of Mathematics at The Hebrew University of Jerusalem.     

Finitely starlike sets whose F-stars have positive measure

Let and assume that there is a countable collection of lines {L _i : 1 i} such that (int cl S) and ((int cl S) S) L _i has one-dimensional Lebesgue measure zero, 1 i. Then every 4 point subset ofS sees viaS a set of positive two-dimensional Lebesgue measure if and only if every finite subset ofS sees viaS such a set. Furthermore, a parallel result holds with two-dimensional replaced by one-dimensional. Finally, setS is finitely starlike if and only if every 5 points ofS see viaS a common point. In each case, the number 4 or 5 is best possible.Supported in part by NSF grant DMS-8705336.     

Entire functions with Julia sets of positive measure

Let f be a transcendental entire function for which the set of critical and asymptotic values is bounded. The Denjoy–Carleman–Ahlfors theorem implies that if the set of all z for which |f(z)| > R has N components for some R > 0, then the order of f is at least N/2. More precisely, we have log log M(r, f) ≥ (N/2) log r − O(1), where M(r, f) denotes the maximum modulus of f. We show that if f does not grow much faster than this, then the escaping set and the Julia set of f have positive Lebesgue measure. However, as soon as the order of f exceeds N/2, this need not be true. The proof requires a sharpened form of an estimate of Carleman and Tsuji related to the Denjoy–Carleman–Ahlfors theorem.     

Paradoxical decompositions of planar sets of positive outer measure

We give examples of paradoxical subsets of the plane which do not have Lebesgue measure zero.     

Pseudocontinuability and properties of analytic functions on boundary sets of positive measure

The following analogue of Fabry's theorem is proved. Assume that a function , analytic in the polydisc, has a sufficiently lacunary Taylor series. If on a subset of the torus, of positive Lebesgue measure, the function coincides in a certain sense with a function analytic in a sufficiently large subset of, then is analytic in the polydisc for some r >1. As a consequence one obtains that a nonconstant function, analytic in a ball and having a sufficiently lacunary Taylor series, cannot have angular boundary values equal in modulus to unity or having zero real part on a set of positive measure.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 126, pp. 88–96, 1983.     

Differentiation of sets in measure

Suppose F(ε), for each ε∈[0,1], is a bounded Borel subset of Rd and F(ε)→F(0) as ε→0. Let A(ε)=F(ε)?F(0) be symmetric difference and P be an absolutely continuous measure on Rd. We introduce the notion of derivative of F(ε) with respect to ε, dF(ε)/dε=dA(ε)/dε, such that

     

On non-enlargeable positive sets

The theory of positive sets on SSD spaces and Banach SNL spaces has been introduced by S. Simons. Monotone sets can be considered as a special case of positive sets. The characterizations of enlargement of positive sets in SSD spaces have been studied by Bo? and Csetnek. In this paper, we present characterizations of non-enlargeable positive sets in SSD spaces and Banach SNL spaces.     

Enlargements of positive sets

In this paper we introduce the notion of enlargement of a positive set in SSD spaces. To a maximally positive set A we associate a family of enlargements E(A) and characterize the smallest and biggest element in this family with respect to the inclusion relation. We also emphasize the existence of a bijection between the subfamily of closed enlargements of E(A) and the family of so-called representative functions of A. We show that the extremal elements of the latter family are two functions recently introduced and studied by Stephen Simons. In this way we extend to SSD spaces some former results given for monotone and maximally monotone sets in Banach spaces.     

The measure of non-normal sets

Summary It is well-known that almost every number in [0, 1] is normal in base 2, in the sense of Lebesgue measure. Kahane and Salem asked whether the same is true with respect to any Borel measure whose Fourier-Stieltjes coefficients vanish at infinity — in other words, whether the set of non-normal numbers is a set of uniqueness in the wide sense. We show that this is not the case. In fact, we give best-possible conditions on the rate of decay of in order that -almost every number be normal. The techniques include, on the one hand, probability measures with respect to which the binary digits in [0, 1] are independent only by blocks, rather than individually, and on the other hand, the strong law of large numbers for weakly correlated random variables.This work was partially supported by an NSF Graduate Fellowship, NSF Grant MCS-82-01602, and an AMS Research Fellowship.     

Construction of sets of positive measure not containing an affine image of a given infinite structure

This paper deals with the problem of existence of infinite structures in euclidean space such that every set of positive measure contains an affine image of it. We contribute to P. Erdös’ question about sequences in the real line, by showing that no triple sum of infinite sets has this property.     

Generalized positive sets and abstract monotonicity

The theory of q-positive sets on SSD spaces has been introduced by Simons (J Convex Anal, 14:297–317, 2007; From Hahn–Banach to monotonicity, Springer, Berlin, 2008). Monotone sets can be considered as special case of q-positive sets. In this paper, we develop a theory of q-positive sets in the framework of abstract monotonicity. We use generalized Fenchel’s duality theorem and give some criteria for maximality of abstract q-positive sets. Finally, we investigate the relation between abstract q-positive sets and abstract convex functions.     

Strongly meager and strong measure zero sets

 In this paper we present two consistency results concerning the existence of large strong measure zero and strongly meager sets. RID="ID=" <E5>Mathematics Subject Classification (2000):</E5>&ensp;03E35 RID="ID=" The first author was supported by Alexander von Humboldt Foundation and NSF grant DMS 95-05375. The second author was partially supported by Basic Research Fund, Israel Academy of Sciences, publication 658 Received: 6 January 1999 / Revised version: 20 July 1999 / Published online: 25 February 2002 RID=" ID=" <E5>Mathematics Subject Classification (2000):</E5>&ensp;03E35 RID=" ID=" The first author was supported by Alexander von Humboldt Foundation and NSF grant DMS 95-05375. The second author was partially supported by Basic Research Fund, Israel Academy of Sciences, publication 658     

Pluripolarity of sets with small Hausdorff measure

We show that any set E⊂C ⁿ , n≥ 2, with finite Hausdorff measure? is pluripolar. The result is sharp with respect to the measuring function. The new idea in the proof is to combine a construction from potential theory, related to the real variational integral , , with properties of the pluricomplex relative extremal function for the Bedford–Taylor capacity. Received: 20 May 1999