Search

Assume two finite families $\mathcal A$ and $\mathcal B$ of convex sets in $\mathbb{R}^3$ have the property that $A\cap B\ne \emptyset$ for every $A \in \mathcal A$ and $B\in \mathcal B$. Is there a constant $\gamma >0$ (independent of $\mathcal A$ and $\mathcal B$) such that there is a line intersecting $\gamma|\mathcal A|$ sets in $\mathcal A$ or $\gamma|\mathcal B|$ sets in $\mathcal B$? This is an intriguing Helly-type question from a paper by Mart\'{i}nez, Roldan and Rubin. We confirm this in the special case when all sets in $\mathcal A$ lie in parallel planes and all sets in $\mathcal B$ lie in parallel planes; in fact, all sets from one of the two families has a line transversal.

This paper is partly an exposition, and partly an extension of our work [1] to the multiparameter case. We consider certain classes of parametrized dynamically defined measures. These are push-forwards, under the natural projection, of ergodic measures for parametrized families of smooth iterated function systems (IFS) on the line. Under some assumptions, most crucially, a transversality condition, we obtain formulas for the Hausdorff dimension of the measure and absolute continuity for almost every parameter in the appropriate parameter region. The main novelty of [1] and the present paper is that not only the IFS, but also the ergodic measure in the symbolic space, whose push-forward we consider, depends on the parameter. This includes many interesting families of measures, in particular, invariant measures for IFS's with place-dependent probabilities and natural (equilibrium) measures for smooth IFS's. One of the goals of this paper is to present an exposition of [1] in a more reader-friendly way, emphasizing the ideas and proof strategies, but omitting the more technical parts. This exposition/survey is based in part on the series of lectures by K\'aroly Simon at the Summer School "Dynamics and Fractals" in 2023 at the Banach Center, Warsaw. The main new feature, compared to [1], is that we consider multi-parameter families; in other words, the set of parameters is allowed to be multi-dimensional. This broadens the scope of applications. A new application considered here is to a class of Furstenberg-like measures. [1] B. B\'ar\'any, K. Simon, B. Solomyak and A. \'Spiewak: Typical absolute continuity for classes of dynamically defined measures. Advances in Mathematics, Volume 399, 2022, 108258, ISSN 0001-8708, https://doi.org/10.1016/j.aim.2022.108258.

The demand for accurate and efficient verification of information in texts generated by large language models (LMs) is at an all-time high, but remains unresolved. Recent efforts have focused on extracting and verifying atomic facts from these texts via prompting LM evaluators. However, we demonstrate that this method of prompting is unreliable when faced with incomplete or inaccurate reference information. We introduce Facts as a Function (FaaF), a new approach to the fact verification task that leverages the function-calling capabilities of LMs. FaaF significantly enhances the ability of LMs to identify unsupported facts in texts, while also improving efficiency and significantly lowering costs compared to prompt-based methods. Additionally, we propose a framework for evaluating factual recall in Retrieval Augmented Generation (RAG) systems, which we employ to compare prompt-based and FaaF methods using various LMs under challenging conditions., Comment: 14 pages, 3 figures

In this paper, we consider a fractal model motivated by the abrasion of convex polyhedra, where the abrasion is realised by chipping small neighbourhoods of vertices. After providing a formal description of the successive chippings, we show that the net of edges converge to a compact limit set under mild assumptions. Furthermore, we study the upper box-counting dimension and the Hausdorff dimension of the limiting object of the net of edges after infinitely many chipping.

We study the level sets of prevalent H\"older functions. For a prevalent $\alpha$-H\"older function on the unit interval, we show that the upper Minkowski dimension of every level set is bounded from above by $1-\alpha$ and Lebesgue positively many level sets have Hausdorff dimension equal to $1-\alpha$., Comment: 14 pages

Given a polytope $P\subset R^3$ and a non-zero vector $z \in R^3$, the plane $\{x\in R^3:zx=t\}$ intersects $P$ in convex polygon $P(z,t)$ for $t \in [t^-,t^+]$ where $t^-=\min \{zx: x \in P\}$ and $t^+=\max \{zx: x\in P\}$, $zx$ is the scalar product of $z,x \in R^3$. Let $A(P,z)$ denote the average number of vertices of $P(z,t)$ on the interval $[t^-,t^+]$. For what polytopes is $A(P,z)$ a constant independent of $z$?

We show that any self-conformal measure $\mu$ on $\mathbb{R}$ is uniformly scaling and generates an ergodic fractal distribution. This generalizes existing results by removing the need for any separation condition. We also obtain applications to the prevalence of normal numbers in self-conformal sets, the resonance between self-conformal measures on the line, and projections of self-affine measures on carpets., Comment: 25 pages

We give a new proof Tverberg's famous theorem: For every set $X \subset \R^d$ with $|X|=(r-1)(d+1)+1$, there is a partition of $X$ into $r$ sets $X_1,\ldots,X_r$ such that $\bigcap_{p=1}^r \conv X_p\ne \emptyset$. The new proof uses linear algebra, specially structured matrices, the theory of linear equations, and Tverberg's original ``moving the points" method.

Assume that $k \le d$ is a positive integer and $\C$ is a finite collection of convex bodies in $\R^d$. We prove a Helly type theorem: If for every subfamily $\C^*\subset \C$ of size at most $\max \{d+1,2(d-k+1)\}$ the set $\bigcap \C^*$ contains a $k$-dimensional cone, then so does $\bigcap \C.$ One ingredient in the proof is another Helly type theorem about the dimension of lineality spaces of convex cones.

The Steinitz lemma, a classic from 1913, states that $a_1,\ldots,a_n$, a sequence of vectors in $\R^d$ with $\sum_1^n a_i=0$, can be rearranged so that every partial sum of the rearranged sequence has norm at most $2d\max \|a_i\|$. In the matrix version $A$ is a $k\times n$ matrix with entries $a_i^j \in \R^d$ with $\sum_{j=1}^k\sum_{i=1}^na_i^j=0$. It is proved in \cite{OPW} that there is a rearrangement of row $j$ of $A$ (for every $j$) such that the sum of the entries in the first $m$ columns of the rearranged matrix has norm at most $40d^5\max \|a_i^j\|$ (for every $m$). We improve this bound to $(4d-2)\max \|a_i^j\|$.

We study the dimension of the attractor and quasi-Bernoulli measures of parametrized families of iterated function systems of non-conformal and non-affine maps. We introduce a transversality condition under which, relying on a weak Ledrappier-Young formula, we show that the dimensions equal to the root of the subadditive pressure and the Lyapunov dimension, respectively, for almost every choice of parameters. We also exhibit concrete examples satisfying the transversality condition with respect to the translation parameters.

We completely characterize the asymptotic behavior of the covering number for strongly connected inhomogeneous graph-directed self-similar sets satisfying the strong open set condition., Comment: 17 pages

We show that the Hausdorff dimension of any slice of the graph of the Takagi function is bounded above by the Assouad dimension of the graph minus one, and that the bound is sharp. The result is deduced from a statement on more general self-affine sets, which is of independent interest. We also prove that the Marstrand's slicing theorem on the graph of the Takagi function extends to all slices if and only if the upper pointwise dimension of every projection of the length measure on the $x$-axis lifted to the graph is at least one., Comment: 37 pages, 5 figures

In this paper, we study the dimension of planar self-affine sets, of which generating iterated function system (IFS) contains non-invertible affine mappings. We show that under a certain separation condition, the dimension equals to the affinity dimension for a typical choice of the linear parts of the non-invertible mappings, furthermore, we show that the dimension is strictly smaller than the affinity dimension for certain choices of parameters.

Two infinite sets $A$ and $B$ of non-negative integers are called \emph{perfect additive complements of non-negative integers}, if every non-negative integer can be uniquely expressed as the sum of elements from $A$ and $B$. In this paper, we define a Lagrange-like spectrum of the perfect additive complements ($\mathfrak{L} $ for short). As a main result, we obtain the smallest accumulation point of the set $\mathfrak{L} $ and prove that the set $\mathfrak{L} $ is closed. Other related results and problems are also contained.

Let $\Sigma_{A}$ be a topologically mixing shift of finite type, let $\sigma:\Sigma_{A}\to\Sigma_{A}$ be the usual left-shift, and let $\mu$ be the Gibbs measure for a H\"{o}lder continuous potential that is not cohomologous to a constant. In this paper we study recurrence rates for the dynamical system $(\Sigma_{A},\sigma)$ that hold $\mu$-almost surely. In particular, given a function $\psi:\mathbb{N}\to \mathbb{N}$ we are interested in the following set $$R_{\psi}=\{{\texttt i}\in \Sigma_{A}:i_{n+1}\ldots i_{n+\psi(n)+1}=i_1\ldots i_{\psi(n)}\textrm{ for infinitely many }n\in\mathbb{N}\}.$$ We provide sufficient conditions for $\mu(R_{\psi})=1$ and sufficient conditions for $\mu(R_{\psi})=0$. As a corollary of these results, we discover a new critical threshold where the measure of $R_{\psi}$ transitions from zero to one. This threshold was previously unknown even in the special case of a non-uniform Bernoulli measure defined on the full shift. The proofs of our results combine ideas from Probability Theory and Thermodynamic Formalism. In our final section we apply our results to the study of dynamics on self-similar sets., Comment: 28 pages. Comments welcome

We consider the minimum number of lines $h_n$ and $p_n$ needed to intersect or pierce, respectively, all the cells of the $n \times n$ chessboard. Determining these values can also be interpreted as a strengthening of the classical plank problem for integer points. Using the symmetric plank theorem of K. Ball, we prove that $h_n = \lceil \frac n 2 \rceil$ for each $n \geq 1$. Studying the piercing problem, we show that $0.7n \leq p_n \leq n-1$ for $n\geq 3$, where the upper bound is conjectured to be sharp. The lower bound is proven by using the linear programming method, whose limitations are also demonstrated., Comment: 15 pages, 7 figures. Final, accepted version. Color of figures modified in order to comply with BW print

One of the most important and useful examples in discrete geometry is a finite sequence of points on the moment curve $\gamma(t)=(t,t^2,t^3,\dots ,t^d)$ or, more generally, on a {\it strictly monotone curve} in $\mathbb R^d$. These sequences as well as the ambient curve itself can be described in terms of {\it universality properties} and we will study the question: "What is a universal sequence of oriented and unoriented lines in $d$-space'' We give partial answers to this question, and to the analogous one for $k$-flats. Given a large integer $n$, it turns out that, like the case of points the number of universal configurations is bounded by a function of $d$, but unlike the case for points, there are a large number of distinct universal finite sequences of lines. We show that their number is at least $2^{d-1}-2$ and at most $(d-1)!$. However, like for points, in all dimensions except $d=4$, there is essentially a unique {\em continuous} example of a universal family of lines. The case $d=4$ is left as an open question., Comment: 21 pages, 2 figures

Extending an earlier result for real matrices we show that multiple eigenvalues of a complex matrix lie in a reduced Gershgorin disk. One consequence is a slightly better estimate in the real case. Another one is a geometric application. Further results of a similar type are given for normal and almost symmetric matrices.

In dynamical systems, shrinking target sets and pointwise recurrent sets are two important classes of dynamically defined subsets. In this article we introduce a mild condition on the linear parts of the affine mappings that allow us to bound the Hausdorff dimension of cylindrical shrinking target and recurrence sets. For generic self-affine sets in the sense of Falconer, that is by randomising the translation part of the affine maps, we prove that these bounds are sharp. These mild assumptions mean that our results significantly extend and complement the existing literature for recurrence on self-affine sets.

Let $\{s_n\}_{n\in\N}$ be a decreasing nonsummable sequence of positive reals. In this paper, we investigate the weighted Birkhoff average $\frac{1}{S_n}\sum_{k=0}^{n-1}s_k\phi(T^kx)$ on aperiodic irreducible subshift of finite type $\Sigma_{\bf A}$ where $\phi: \Sigma_{\bf A}\mapsto \R$ is a continuous potential. Firstly, we show the entropy spectrum of the weighed Birkhoff averages remains the same as that of the Birkhoff averages. Then we calculate the packing spectrum of the weighed Birkhoff averages. It turns out that we can have two cases, either the packing dimension of every level set equals to its Hausdorff dimension or for every nonempty level set it is equal to the packing dimension of the whole space.

In this paper, we present a variety of problems in the interface between combinatorics and geometry around the theorems of Helly, Radon, Carath\'eodory, and Tverberg. Through these problems we describe the fascinating area of Helly-type theorems, and explain some of its main themes and goals.

It is generally agreed upon that most individuals who acquire a college degree do so in their early 20s. Despite this consensus, we show that in the US from the 1930 birth cohort onwards a large fraction -- around 20% -- of college graduates obtained their degree after age 30. We explore the implications of this phenomenon. First, we show that these so-called late bloomers have significantly contributed to the narrowing of gender and racial gaps in the college share, despite the general widening of the racial gap. Second, late bloomers are responsible for more than half of the increase in the aggregate college share from 1960 onwards. Finally, we show that the returns to having a college degree vary depending on the age at graduation. Ignoring the existence of late bloomers therefore leads to a significant underestimation of the returns to college education for those finishing college in their early 20s.

We consider one-parameter families of smooth uniformly contractive iterated function systems $\{f^\lambda_j\}$ on the real line. Given a family of parameter dependent measures $\{\mu_{\lambda}\}$ on the symbolic space, we study geometric and dimensional properties of their images under the natural projection maps $\Pi^\lambda$. The main novelty of our work is that the measures $\mu_\lambda$ depend on the parameter, whereas up till now it has been usually assumed that the measure on the symbolic space is fixed and the parameter dependence comes only from the natural projection. This is especially the case in the question of absolute continuity of the projected measure $(\Pi^\lambda)_*\mu_\lambda$, where we had to develop a new approach in place of earlier attempt which contains an error. Our main result states that if $\mu_\lambda$ are Gibbs measures for a family of H\"older continuous potentials $\phi^\lambda$, with H\"older continuous dependence on $\lambda$ and $\{\Pi^\lambda\}$ satisfy the transversality condition, then the projected measure $(\Pi^\lambda)_*\mu_\lambda$ is absolutely continuous for Lebesgue a.e.\ $\lambda$, such that the ratio of entropy over the Lyapunov exponent is strictly greater than $1$. We deduce it from a more general almost sure lower bound on the Sobolev dimension for families of measures with regular enough dependence on the parameter. Under less restrictive assumptions, we also obtain an almost sure formula for the Hausdorff dimension. As applications of our results, we study stationary measures for iterated function systems with place-dependent probabilities (place-dependent Bernoulli convolutions and the Blackwell measure for binary channel) and equilibrium measures for hyperbolic IFS with overlaps (in particular: natural measures for non-homogeneous self-similar IFS and certain systems corresponding to random continued fractions).

For a planar self-affine set satisfying the strong separation condition, it has been recently proved that under mild assumptions the Hausdorff dimension equals the affinity dimension. In this article, we continue this line of research and our objective is to acquire more refined geometric information. In a large class of non-carpet planar self-affine sets, we characterize Ahlfors regularity, determine the Assouad dimension of the set and its projections, and estimate the Hausdorff dimension of slices. We also demonstrate that the Assouad dimension is not necessarily bounded above by the affinity dimension., Comment: 48 pages, 2 figures

We prove that given a finite collection of cylinders in $\R^3$ with the property that any two them intersect, then there is a line intersecting an $\alpha$ fraction of the cylinders where $\alpha=\frac 1{28}$. This is a special case of an interesting conjecture., Comment: 6 pages

We extend the famous Erd\H{o}s-Szekeres theorem to $k$-flats in ${\mathbb{R}^d}$

This paper studies how long it takes the orbit of the chaos game to reach a certain density inside the attractor of a strictly contracting iterated function system of which we only assume that its lower dimension is positive. We show that the rate of growth of this cover time is determined by the Minkowski dimension of the push-forward of the shift invariant measure with exponential decay of correlations driving the chaos game. Moreover, we bound the expected value of the cover time from above and below with multiplicative logarithmic correction terms. As an application, for Bedford-McMullen carpets we completely characterise the family of probability vectors which minimise the Minkowski dimension of Bernoulli measures. Interestingly, these vectors have not appeared in any other aspect of Bedford-McMullen carpets before., Comment: 29 pages, 3 figures, 2 tables

We revisit the cocked hat -- an old problem from navigation -- and examine under what conditions its old solution is valid., Comment: 13 pages, 8 figures

In this paper, we study the topological spectrum of weighted Birkhoff averages over aperiodic and irreducible subshifts of finite type. We show that for a uniformly continuous family of potentials, the spectrum is continuous and concave over its domain. In case of typical weights with respect to some ergodic quasi-Bernoulli measure, we determine the spectrum. Moreover, in case of full shift and under the assumption that the potentials depend only on the first coordinate, we show that our result is applicable for regular weights, like M\"obius sequence.

This paper considers self-conformal iterated function systems (IFSs) on the real line whose first level cylinders overlap. In the space of self-conformal IFSs, we show that generically (in topological sense) if the attractor of such a system has Hausdorff dimension less than $1$ then it has zero appropriate dimensional Hausdorff measure and its Assouad dimension is equal to $1$. Our main contribution is in showing that if the cylinders intersect then the IFS generically does not satisfy the weak separation property and hence, we may apply a recent result of Angelevska, K\"aenm\"aki and Troscheit [BLMS, 2020]. This phenomenon holds for transversal families (in particular for the translation family) typically, in the self-similar case, in both topological and in measure theoretical sense, and in the more general self-conformal case in the topological sense., Comment: 27 pages including Appendix; v2: minor updates, accepted version in RSE Proc A

The Guarded Negation Fragment (GNFO) is a fragment of first-order logic that contains all positive existential formulas, can express the first-order translations of basic modal logic and of many description logics, along with many sentences that arise in databases. It has been shown that the syntax of GNFO is restrictive enough so that computational problems such as validity and satisfiability are still decidable. This suggests that, in spite of its expressive power, GNFO formulas are amenable to novel optimizations. In this paper we study the model theory of GNFO formulas. Our results include effective preservation theorems for GNFO, effective Craig Interpolation and Beth Definability results, and the ability to express the certain answers of queries with respect to a large class of GNFO sentences within very restricted logics. This version of the paper contains streamlined and corrected versions of results concerning entailment of a conjunctive query from a set of ground facts and a theory consisting of GNFO sentences of a special form ("dependencies").

It is well-known that a line can intersect at most $2n-1$ cells of the $n \times n$ chessboard. Here we consider the high dimensional version: how many cells of the $d$-dimensional $n\times \ldots \times n$ box can a hyperplane intersect? We also prove the lattice analogue of the following well-known fact. If $K,L$ are convex bodies in $R^d$ and $K\subset L$, then the surface area of $K$ is smaller than that of $L$.

In this article, we study the Hausdorff measure of shrinking target sets on self-conformal sets. The Hausdorff dimension of the sets we are interested in here was established by Hill and Velani in 1995. However, until recently, little more was known about the Hausdorff measure of these particular sets. In this paper we provide a complete characterisation of the Hausdorff measure of these sets, obtaining a dichotomy for the Hausdorff measure which is determined by the convergence or divergence of a sum depending on the radii of our "shrinking targets". Our main result complements earlier work of Levesley, Salp, and Velani~(2007), and recent work of Baker (2019)., Comment: 33 pages