A set $S \subseteq \mathbb{R}$ is almost Erdős if, for every $\varepsilon \gt 0$, there exists a set $E \subseteq \mathbb{R}$ of positive Lebesgue measure such that $\{x \in S : ax+b \notin E\}$ is nonempty for all $|a| \gt \varepsilon$ and $b \in \mathbb{R}$. In this note, we show that any decreasing null sequence $(x_n)$ with decay rate greater than $1/2$ is an almost Erdős set.

We consider $L^q$-spectra of planar graph-directed self-affine measures generated by diagonal or anti-diagonal matrices. Assuming the directed graph is strongly connected and the system satisfies the rectangular open set condition, we obtain a general closed form expression for the $L^q$-spectra. Consequently, we obtain a closed form expression for box dimensions of associated planar graph-directed box-like self-affine sets. We also provide a precise answer to a question posed by Fraser [On the $L^q$-spectrum of planar self-affine measures. Trans. Amer. Math. Soc. 368(8) (2016), 5579–5620] concerning the $L^q$-spectra of planar self-affine measures generated by diagonal matrices. An interesting observation of the closed form expression is that it is possible to calculate the $L^q$-spectrum of a measure without involving the exact $L^q$-spectra of its projections to the axes.

This article focuses on the occurrence of 3-point configurations in subsets of $\mathbb {R}^d$ of sufficient thickness. We prove that a compact set $A\subset \mathbb {R}^d$ contains a similar copy of any linear $3$-point configuration (such as a $3$-point arithmetic progression) provided that A satisfies a mild Yavicoli-thickness condition and an r-uniformity condition for $d\geq 2$; or, when $d=1$, the result holds provided that the Newhouse thickness of A is at least $1$. Moreover, we prove that compact sets $A\subset \mathbb {R}^2$ contain the vertices of an equilateral triangle (and more generally, the vertices of a similar copy of any given triangle) provided A satisfies a mild Yavicoli-thickness condition and an r-uniformity condition. Further, $C\times C$ contains the vertices of an equilateral triangle (and more generally the vertices of a similar copy of any given 3-point configuration) provided the Newhouse thickness of C is at least $1$. These are among the first results in the literature to give explicit criteria for the occurrence of 3-point configurations in the plane.

Let X be an uncountable Polish space and let $\mathcal {I}$ be an ideal on $\omega $. A point $\eta \in X$ is an $\mathcal {I}$-limit point of a sequence $(x_n)$ taking values in X if there exists a subsequence $(x_{k_n})$ convergent to $\eta $ such that the set of indexes $\{k_n: n \in \omega \}\notin \mathcal {I}$. Denote by $\mathscr {L}(\mathcal {I})$ the family of subsets $S\subseteq X$ such that S is the set of $\mathcal {I}$-limit points of some sequence taking values in X or S is empty. In this article, we study the relationships between the topological complexity of ideals $\mathcal {I}$, their combinatorial properties, and the families of sets $\mathscr {L}(\mathcal {I})$ which can be attained. On the positive side, we provide several purely combinatorial (not depending on the space X) characterizations of ideals $\mathcal {I}$ for the inclusions and the equalities between $\mathscr {L}(\mathcal {I})$ and the Borel classes $\Pi ^0_1$, $\Sigma ^0_2$, and $\Pi ^0_3$. As a consequence, we prove that if $\mathcal {I}$ is a $\Pi ^0_4$ ideal then exactly one of the following cases holds: $\mathscr {L}(\mathcal {I})=\Pi ^0_1$ or $\mathscr {L}(\mathcal {I})=\Sigma ^0_2$ or $\mathscr {L}(\mathcal {I})=\Sigma ^1_1$ (however we do not have an example of a $\Pi ^0_4$ ideal with $\mathscr {L}(\mathcal {I})=\Sigma ^1_1$). In addition, we provide an explicit example of a coanalytic ideal $\mathcal {I}$ for which $\mathscr {L}(\mathcal {I})=\Sigma ^1_1$. On the negative side, since $\mathscr {L}(\mathcal {I})$ contains all singletons, it is immediate that there are no ideals $\mathcal {I}$ such that $\mathscr {L}(\mathcal {I})=\Sigma ^0_1$. On the same direction, we show that there are no ideals $\mathcal {I}$ such that $\mathscr {L}(\mathcal {I})=\Pi ^0_2$ or $\mathscr {L}(\mathcal {I})=\Sigma ^0_3$. In fact, for instance, if $\mathcal {I}$ is a Borel ideal and $\mathscr {L}(\mathcal {I})$ contains a non $\Sigma ^0_2$ set, then it contains all $\Pi ^0_3$ sets. We conclude with several open questions.

We study the sets of points where a Lévy function and a translated Lévy function share a given couple of Hölder exponents, and we investigate how their Hausdorff dimensions depend on the translation parameter.

In this paper we prove disintegration results for self-conformal measures and affinely irreducible self-similar measures. The measures appearing in the disintegration resemble self-conformal/self-similar measures for iterated function systems satisfying the strong separation condition. We use these disintegration statements to prove new results on the Diophantine properties of these measures.

Let $\gamma _{n}= O (\log ^{-c}n)$ and let $\nu $ be the infinite product measure whose nth marginal is Bernoulli $(1/2+\gamma _{n})$. We show that $c=1/2$ is the threshold, above which $\nu $-almost every point is simply Poisson generic in the sense of Peres and Weiss, and below which this can fail. This provides a range in which $\nu $ is singular with respect to the uniform product measure, but $\nu $-almost every point is simply Poisson generic.

Given $\beta>1$, let $T_\beta $ be the $\beta $-transformation on the unit circle $[0,1)$, defined by $T_\beta (x)=\beta x-\lfloor \beta x\rfloor $. For each $t\in [0,1)$, let $K_\beta (t)$ be the survivor set consisting of all $x\in [0,1)$ whose orbit $\{T^n_\beta (x): n\ge 0\}$ never enters the interval $[0,t)$. Kalle et al [Ergod. Th. & Dynam. Sys. 40(9) (2020), 2482–2514] considered the case $\beta \in (1,2]$. They studied the set-valued bifurcation set $\mathscr {E}_\beta :=\{t\in [0,1): K_\beta (t')\ne K_\beta (t)~\text { for all } t'>t\}$ and proved that the Hausdorff dimension function $t\mapsto \dim _H K_\beta (t)$ is a non-increasing Devil’s staircase. In a previous paper [Ergod. Th. & Dynam. Sys. 43(6) (2023), 1785–1828], we determined, for all $\beta \in (1,2]$, the critical value $\tau (\beta ):=\min \{t>0: \eta _\beta (t)=0\}$. The purpose of the present article is to extend these results to all $\beta>1$. In addition to calculating $\tau (\beta )$, we show that: (i) the function $\tau : \beta \mapsto \tau (\beta )$ is left-continuous on $(1,\infty )$ with right-hand limits everywhere, but has countably infinitely many discontinuities; (ii) $\tau $ has no downward jumps; and (iii) there exists an open set $O\subset (1,\infty )$, whose complement $(1,\infty )\setminus O$ has zero Hausdorff dimension, such that $\tau $ is real-analytic, strictly convex, and strictly decreasing on each connected component of O. We also prove several topological properties of the bifurcation set $\mathscr {E}_\beta $. The key to extending the results from $\beta \in (1,2]$ to all $\beta>1$ is an appropriate generalization of the Farey words that are used to parameterize the connected components of the set O. Some of the original proofs from the above-mentioned papers are simplified.

Paul Erdős and R. Daniel Mauldin asked a series of questions on certain types of polygons of area $1$, the vertices of which can be found in every planar set of infinite Lebesgue measure. We address two of these questions, one on cyclic quadrilaterals and the other on convex polygons with congruent sides, with, respectively, positive and negative answers.

In recent years, there has been significant interest in the effect of different types of adversarial perturbations in data classification problems. Many of these models incorporate the adversarial power, which is an important parameter with an associated trade-off between accuracy and robustness. This work considers a general framework for adversarially perturbed classification problems, in a large data or population-level limit. In such a regime, we demonstrate that as adversarial strength goes to zero that optimal classifiers converge to the Bayes classifier in the Hausdorff distance. This significantly strengthens previous results, which generally focus on $L^1$-type convergence. The main argument relies upon direct geometric comparisons and is inspired by techniques from geometric measure theory.

We construct large Salem sets avoiding patterns, complementing previous constructions of pattern avoiding sets with large Hausdorff dimension. For a (possibly uncountable) family of uniformly Lipschitz functions $\{f_i:(\mathbb{T}^d)^{n-2} \to \mathbb{T}^d \}$, we obtain a Salem subset of $\mathbb{T}^d$ with dimension $d/(n-1)$ avoiding nontrivial solutions to the equation $x_n-x_{n-1} = f_i(x_1,...,x_{n-2})$. For a countable family of smooth functions $\{f_i: (\mathbb{T}^d)^{n-1} \to \mathbb{T}^d \}$ satisfying a modest geometric condition, we obtain a Salem subset of $\mathbb{T}^d$ with dimension $d/(n-3/4)$ avoiding nontrivial solutions to the equation $x_n= f(x_1,...,x_{n-1})$. For a set $Z \subset \mathbb{T}^{dn}$ which is the countable union of a family of sets, each with lower Minkowski dimension $s$, we obtain a Salem subset of $\mathbb{T}^d$ of dimension $(dn-s)/(n-1/2)$ whose Cartesian product does not intersect $Z$ except at points with non-distinct coordinates.

In deep learning (DL), the instability phenomenon is widespread and well documented, and the most commonly used measure of stability is the Lipschitz constant. While a small Lipchitz constant is traditionally viewed as guarantying stability, it does not capture the instability phenomenon in DL for classification well. The reason is that a classification function – which is the target function to be approximated – is necessarily discontinuous, thus having an ‘infinite’ Lipchitz constant. As a result, the classical approach will deem every classification function unstable, yet basic classification functions a la ‘is there a cat in the image?’ will typically be locally very ‘flat’ – and thus locally stable – except at the decision boundary. The lack of an appropriate measure of stability hinders a rigorous theory for stability in DL, and consequently, there are no proper approximation theoretic results that can guarantee the existence of stable networks for classification functions. In this paper, we introduce a novel stability measure $\mathcal{S}(f)$, for any classification function $f$, appropriate to study the stability of classification functions and their approximations. We further prove two approximation theorems: First, for any $\epsilon \gt 0$ and any classification function $f$ on a compact set, there is a neural network (NN) $\psi$, such that $\psi - f \neq 0$ only on a set of measure $\lt \epsilon$; moreover, $\mathcal{S}(\psi ) \geq \mathcal{S}(f) - \epsilon$ (as accurate and stable as $f$ up to $\epsilon$). Second, for any classification function $f$ and $\epsilon \gt 0$, there exists a NN $\psi$ such that $\psi = f$ on the set of points that are at least $\epsilon$ away from the decision boundary.

We extend the theoretical framework of proof mining by establishing general logical metatheorems that allow for the extraction of the computational content of theorems with prima facie “noncomputational” proofs from probability theory, thereby unlocking a major branch of mathematics as a new area of application for these methods. Concretely, we first devise proof-theoretically tame logical systems that allow for the formalization of proofs involving algebras of sets together with probability contents, that is probability measures which are only assumed to be finitely additive. Based on these systems, we provide extensions for the tame treatment of Lebesgue integrals on probability contents as well as $\sigma $-algebras and associated probability measures, all via intensional approaches. All these systems are then shown to be amenable to proof-theoretic metatheorems in the style of proof mining which guarantee the extractability of effective and tame bounds from large classes of ineffective existence proofs in probability theory. Moreover, these extractable bounds are guaranteed to be highly uniform in the sense that they will be independent of all parameters relating to the underlying probability space, particularly regarding events or measures of them. As such, these results in particular provide the first logical explanation for the success and the observed uniformities of the previous ad hoc case studies of proof mining in these areas and further illustrate their extent. Lastly, we establish a general proof-theoretic transfer principle that allows for the lift of quantitative information on a relationship between different modes of convergence for sequences of real numbers to sequences of random variables.

We characterize the subsets E of a metric space X with doubling measure whose distance function to some negative power $\operatorname{dist}(\cdot,E)^{-\alpha}$ belongs to the Muckenhoupt A1 class of weights in X. To this end, we introduce the weakly porous sets in this setting, and show that, along with certain doubling-type conditions for the sizes of the largest E-free holes, these sets characterize the mentioned A1-property. We exhibit examples showing the optimality of these conditions, and simplify them in the particular case where the underlying measure satisfies a qualitative annular decay property. In addition, we use some of these distance functions as a new and simple method to explicitly construct doubling weights in ${\mathbb R}^n$ that do not belong to $A_\infty.$

In this paper, we investigate the dimension theory of the one-parameter family of Okamoto’s function. We compute the Hausdorff, box-counting, and Assouad dimensions of the graph for a typical choice of parameter. Furthermore, we study the dimension of the level sets. We give an upper bound on the dimension of every level set, and we show that for a typical choice of parameter, this value is attained for Lebesgue almost every level set.

Recently, Barros et al. [‘On the shortest distance between orbits and the longest common substring problem’, Adv. Math. 344 (2019), 311–339] adopted a dynamical system perspective to study the decay of the shortest distance between orbits. We calculate the Hausdorff dimensions of the exceptional sets arising from the shortest distance between orbits in conformal iterated function systems.

We investigate the possibility of defining meaningful upper and lower quantization dimensions for a compactly supported Borel probability measure of order r, including negative values of r. To this end, we employ the concept of partition functions, which generalises the notion of the $L^q$-spectrum, thus extending the authors’ earlier work with Sanguo Zhu in a natural way. In particular, we derive inherent fractal-geometric bounds and easily verifiable necessary conditions for the existence of quantization dimensions. We state the exact asymptotics of the quantization error of negative order for absolutely continuous measures, thereby providing an affirmative answer to an open question regarding the geometric mean error posed by Graf and Luschgy in this journal in 2004.

Let $[a_1(x),a_2(x),a_3(x),\dots ]$ be the continued fraction expansion of an irrational number $x\in (0,1)$. Denote by $S_{n}(x):=\sum _{k=1}^{n} a_{k}(x)$ the sum of partial quotients of x. From the results of Khintchine (1935), Diamond and Vaaler (1986), and Philipp (1988), it follows that for Lebesgue almost every $x \in (0,1)$,

$$\begin{align*}\liminf _{n \rightarrow \infty} \frac{S_{n}(x)}{n \log n}=\frac{1}{\log 2} \quad \text {and} \quad \limsup _{n \rightarrow \infty} \frac{S_{n}(x)}{n \log n}=\infty. \end{align*}$$

We establish the pointwise equidistribution of self-similar measures in the complex plane. Let $\beta \in \mathbb Z[\mathrm{i}]$, whose complex conjugate $\overline{\beta}$ is not a divisor of β, and $T \subset \mathbb Z[\mathrm{i}]$ a finite subset. Let µ be a non-atomic self-similar measure with respect to the IFS $\big\{f_{t}(z)=\frac{z+t}{\beta}\colon t\in T\big\}$. For $\alpha \in \mathbb Z[\mathrm{i}]$, if α and β are relatively prime, then we show that the sequence $(\alpha^n z)_{n\ge 1}$ is equidistributed modulo one for µ-almost everywhere $z \in \mathbb{C}$. We also discuss normality of radix expansions in Gaussian integer base, and obtain pointwise normality. Our results generalize partially the classical results in the real line to the complex plane.

Let $ K $ be a compact subset of the d-torus invariant under an expanding diagonal endomorphism with s distinct eigenvalues. Suppose the symbolic coding of K satisfies weak specification. When $ s \leq 2 $, we prove that the following three statements are equivalent: (A) the Hausdorff and box dimensions of $ K $ coincide; (B) with respect to some gauge function, the Hausdorff measure of $ K $ is positive and finite; (C) the Hausdorff dimension of the measure of maximal entropy on $ K $ attains the Hausdorff dimension of $ K $. When $ s \geq 3 $, we find some examples in which statement (A) does not hold but statement (C) holds, which is a new phenomenon not appearing in the planar cases. Through a different probabilistic approach, we establish the equivalence of statements (A) and (B) for Bedford–McMullen sponges.