arXiv subjects

Tamás Keleti

Publications and source records attributed to Tamás Keleti.

26 records · Page 2Linked to original sources

arXiv2012

How large dimension guarantees a given angle?

We study the following two problems: (1) Given $n\ge 2$ and $\al$, how large Hausdorff dimension can a compact set $A\su\Rn$ have if $A$ does not contain three points that form an angle $\al$? (2) Given $\al$ and $\de$, how large Hausdorff dimension can a %compact subset $A$ of a Euclidean space have if $A$ does not contain three points that form an angle in the $\de$-neighborhood of $\al$? An interesting phenomenon is that different angles show different behaviour in the above problems. Apart from the clearly special extreme angles 0 and $180^\circ$, the angles $60^\circ,90^\circ$ and $120^\circ$ also play special role in problem (2): the maximal dimension is smaller for these special angles than for the other angles. In problem (1) the angle $90^\circ$ seems to behave differently from other angles.

math.CA↗

arXiv2012

Hausdorff dimension and non-degenerate families of projections

We study parametrized families of orthogonal projections for which the dimension of the parameter space is strictly less than that of the Grassmann manifold. We answer the natural question of how much the Hausdorff dimension may decrease by verifying the best possible lower bound for the dimension of almost all projections of a finite measure. We also show that a similar result is valid for smooth families of maps from $n$-dimensional Euclidean space to $m$-dimensional one.

math.CA↗

arXiv2012

Reconstructing geometric objects from the measures of their intersections with test sets

Let us say that an element of a given family $\A$ of subsets of $\R^d$ can be reconstructed using $n$ test sets if there exist $T_1,...,T_n \subset \R^d$ such that whenever $A,B\in \A$ and the Lebesgue measures of $A \cap T_i$ and $B \cap T_i$ agree for each $i=1,...,n$ then $A=B$. Our goal will be to find the least such $n$. We prove that if $\A$ consists of the translates of a fixed reasonably nice subset of $\R^d$ then this minimum is $n=d$. In order to obtain this result we reconstruct a translate of a fixed function using $d$ test sets as well, and also prove that under rather mild conditions the measure function $f_{K,θ} (r) = \la^{d-1} (K \cap \{x \in \RR^d : = r\})$ of the sections of $K$ is absolutely continuous for almost every direction $θ$. These proofs are based on techniques of harmonic analysis. We also show that if $\A$ consists of the magnified copies $rE+t$ $(r\ge 1, t\in\R^d)$ of a fixed reasonably nice set $E\subset \R^d$, where $d\ge 2$, then $d+1$ test sets reconstruct an element of $\A$. This fails in $\R$: we prove that an interval, and even an interval of length at least 1 cannot be reconstructed using 2 test sets. Finally, using randomly constructed test sets, we prove that an element of a reasonably nice $k$-dimensional family of geometric objects can be reconstructed using $2k+1$ test sets. A example from algebraic topology shows that $2k+1$ is sharp in general.

math.CA↗

arXiv2011

The Fixed Point of the Composition of Derivatives

We give an affirmative answer to a question of K. Ciesielski by showing that the composition $f\circ g$ of two derivatives $f,g:[0,1]\to[0,1]$ always has a fixed point. Using Maximoff's Theorem we obtain that the composition of two $[0,1]\to[0,1]$ Darboux Baire-1 functions must also have a fixed point.

math.CA↗

arXiv2011

Is Lebesgue measure the only $σ$-finite invariant Borel measure?

R.D.Mauldin asked if every translation invariant $σ$-finite Borel measure on $\RR^d$ is a constant multiple of Lebesgue measure. The aim of this paper is to show that the answer is "yes and no", since surprisingly the answer depends on what we mean by Borel measure and by constant. We present Mauldin's proof of what he called a folklore result, stating that if the measure is only defined for Borel sets then the answer is affirmative. Then we show that if the measure is defined on a $σ$-algebra \emph{containing} the Borel sets then the answer is negative. However, if we allow the multiplicative constant to be infinity, then the answer is affirmative in this case as well. Moreover, our construction also shows that an isometry invariant $σ$-finite Borel measure (in the wider sense) on $\RR^d$ can be non-$σ$-finite when we restrict it to the Borel sets.

math.CA↗

arXiv2011

Borel sets which are null or non-$σ$-finite for every translation invariant measure

We show that the set of Liouville numbers is either null or non-$σ$-finite with respect to every translation invariant Borel measure on $\RR$, in particular, with respect to every Hausdorff measure $\iH^g$ with gauge function $g$. This answers a question of D. Mauldin. We also show that some other simply defined Borel sets like non-normal or some Besicovitch-Eggleston numbers, as well as all Borel subgroups of $\RR$ that are not $F_σ$ possess the above property. We prove that, apart from some trivial cases, the Borel class, Hausdorff or packing dimension of a Borel set with no such measure on it can be arbitrary.

math.CA↗

arXiv2008

Self-similar and self-affine sets; measure of the intersection of two copies

Let K be a self-similar or self-affine set in R^d, let μbe a self-similar or self-affine measure on it, and let G be the group of affine maps, similitudes, isometries or translations of R^d. Under various assumptions (such as separation conditions or we assume that the transformations are small perturbations or that K is a so called Sierpinski sponge) we prove theorems of the following types, which are closely related to each other; Non-stability: There exists a constant c<1 such that for every g\in G we have either μ(K\cap g(K)) 0 \iff int_K (K\cap g(K)) is nonempty (where int_K is interior relative to K). Extension: The measure μhas a G-invariant extension to R^d. Moreover, in many situations we characterize those g's for which μ(K\cap g(K) > 0, and we also get results about those $g$'s for which $g(K)\su K$ or $g(K)\supset K$ holds.

math.GM↗

arXiv2007

Invariant decomposition of functions with respect to commuting invertible transformations

Consider a_1,a_2,...,a_n, arbitrary elements of R. We characterize those real functions f that decompose into the sum of a_j-periodic functions, i.e., f=f_1+...+f_n with D_{a_j}f(x):=f(x+a_j)-f(x)=0. We show that f has such a decomposition if and only if for all partitions to B_1, B_2,... B_N of {a_1,a_2,...,a_n} with B_j consisting of commensurable elements with least common multiples b_j, one has D_{b_1}... D_{b_N}f=0. Actually, we prove a more general result for periodic decompositions of real functions f defined on an Abelian group A, and, in fact, we even consider invariant decompositions of functions f defined on some abstract set A, with respect to commuting, invertible self-mappings of the set A. We also extend our results to functions between torsion free Abelian groups. As a corollary we also obtain that on a torsion free Abelian group the existence of a real valued periodic decomposition of an integer valued function implies the existence of an integer valued periodic decomposition with the same periods.

math.CA↗