Are lines much bigger than line segments?

We pose the following conjecture: (*) If A is the union of line segments in R^n, and B is the union of the corresponding full lines then the Hausdorff dimensions of A and B agree. We prove that this conjecture would imply that every Besicovitch set (compact set that contains line segments in every direction) in R^n has Hausdorff dimension at least n-1 and (upper) Minkowski dimension n. We also prove that conjecture (*) holds if the Hausdorff dimension of B is at most 2, so in particular it holds in the plane.Comment: minor corrections, "much" was added in the titl

Is Lebesgue measure the only σ\sigma-finite invariant Borel measure?

R.D.Mauldin asked if every translation invariant $\sigma$-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 $\sigma$-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 $\sigma$-finite Borel measure (in the wider sense) on \RR^d can be non-$\sigma$-finite when we restrict it to the Borel sets

On the determination of sets by their triple correlation in finite cyclic groups

Let $G$ be a finite abelian group and $E$ a subset of it. Suppose that we know for all subsets $T$ of $G$ of size up to $k$ for how many $x \in G$ the translate $x+T$ is contained in $E$. This information is collectively called the $k$-deck of $E$. One can naturally extend the domain of definition of the $k$-deck to include functions on $G$. Given the group $G$ when is the $k$-deck of a set in $G$ sufficient to determine the set up to translation? The 2-deck is not sufficient (even when we allow for reflection of the set, which does not change the 2-deck) and the first interesting case is $k=3$. We further restrict $G$ to be cyclic and determine the values of $n$ for which the 3-deck of a subset of \ZZ_n is sufficient to determine the set up to translation. This completes the work begun by Gr\"unbaum and Moore as far as the 3-deck is concerned. We additionally estimate from above the probability that for a random subset of \ZZ_n there exists another subset, not a translate of the first, with the same 3-deck. We give an exponentially small upper bound when the previously known one was $O(1\bigl / \sqrt{n})$.Comment: 17 page

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

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,\theta} (r) = \la^{d-1} (K \cap \{x \in \RR^d : = r\}) of the sections of $K$ is absolutely continuous for almost every direction $\theta$. 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

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 \mu be 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 \mu(K\cap g(K)) <c \mu(K) or K\subset g(K). Measure and topology: For every g\in G we have \mu(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 \mu has a G-invariant extension to R^d. Moreover, in many situations we characterize those g's for which \mu(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

Squares and their centers

We study the relationship between the sizes of two sets $B, S\subset\mathbb{R}^2$ when $B$ contains either the whole boundary, or the four vertices, of a square with axes-parallel sides and center in every point of $S$, where size refers to one of cardinality, Hausdorff dimension, packing dimension, or upper or lower box dimension. Perhaps surprinsingly, the results vary depending on the notion of size under consideration. For example, we construct a compact set $B$ of Hausdorff dimension $1$ which contains the boundary of an axes-parallel square with center in every point $[0,1]^2$, but prove that such a $B$ must have packing and lower box dimension at least $\tfrac{7}{4}$, and show by example that this is sharp. For more general sets of centers, the answers for packing and box counting dimensions also differ. These problems are inspired by the analogous problems for circles that were investigated by Bourgain, Marstrand and Wolff, among others.Comment: 20 pages, no figure

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

Better bounds for planar sets avoiding unit distances

A $1$-avoiding set is a subset of $\mathbb{R}^n$ that does not contain pairs of points at distance $1$. Let $m_1(\mathbb{R}^n)$ denote the maximum fraction of $\mathbb{R}^n$ that can be covered by a measurable $1$-avoiding set. We prove two results. First, we show that any $1$-avoiding set in $\mathbb{R}^n$ ($n\ge 2$) that displays block structure (i.e., is made up of blocks such that the distance between any two points from the same block is less than $1$ and points from distinct blocks lie farther than $1$ unit of distance apart from each other) has density strictly less than $1/2^n$. For the special case of sets with block structure this proves a conjecture of Erd\H{o}s asserting that $m_1(\mathbb{R}^2) < 1/4$. Second, we use linear programming and harmonic analysis to show that $m_1(\mathbb{R}^2) \leq 0.258795$.Comment: 16 pages, 1 figure. Contains a Sage script called dstverify.sage, to verify the application of Theorem 3.3. Download the article source to get the scrip