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We obtain a criterion for an analytic subset of a Euclidean space to contain points of differentiability of a typical Lipschitz function, namely, that it cannot be covered by countably many sets, each of which is closed and purely…

The Erd\H os unit distance conjecture in the plane says that the number of pairs of points from a point set of size $n$ separated by a fixed (Euclidean) distance is $\leq C_{\epsilon} n^{1+\epsilon}$ for any $\epsilon>0$. The best known…

For a separable finite diffuse measure space $\mathcal{M}$ and an orthonormal basis $\{\varphi_n\}$ of $L^2(\mathcal{M})$ consisting of bounded functions $\varphi_n\in L^\infty(\mathcal{M})$, we find a measurable subset…

In this paper we start to investigate a new body of questions in additive combinatorics. The fundamental Cauchy--Davenport theorem gives a lower bound on the size of a sumset A+B for subsets of the cyclic group Zp of order p (p prime), and…

We establish the Level-1 and Level-3 Large Deviation Principles (LDPs) for invariant measures on shift spaces over finite alphabets under very general decoupling conditions for which the thermodynamic formalism does not apply. Such…

Left on its own, a quantum state evolves deterministically under the Schr\"odinger Equation, forming superpositions. Upon measurement, however, a stochastic process governed by the Born rule collapses it to a single outcome. This dual…

Using a characterization of Mutual Complete Dependence copulas, we show that, with respect to the Sobolev norm, the MCD copulas can be approximated arbitrarily closed by shuffles of Min. This result is then used to obtain a characterization…

In this paper, we study an extension of the CPE conjecture to manifolds $M$ which support a structure relating curvature to the geometry of a smooth map $\varphi : M \to N$. The resulting system, denoted by $(\varphi-\mathrm{CPE})$, is…

A symmetric subset of the reals is one that remains invariant under some reflection x --> c-x. Given 0 < x < 1, there exists a real number D(x) with the following property: if 0 < d < D(x), then every subset of [0,1] with measure x contains…

We establish the following max-plus analogue of Minkowski's theorem. Any point of a compact max-plus convex subset of $(R\cup\{-\infty\})^n$ can be written as the max-plus convex combination of at most $n+1$ of the extreme points of this…

Let $S$ be a family of sequences of positive numbers that decrease to 0, let $X$ be a metric space and $A \subset X$. $A$ is said to be $S$-dominated if, for every $s\in S$, a countable cover $\{E_n\}$ of $E$ can be found such that $diam…

Measuring the dependence of data plays a central role in statistics and machine learning. In this work, we summarize and generalize the main idea of existing information-theoretic dependence measures into a higher-level perspective by the…

We introduce a new notion of a harmonic measure for a $d$-dimensional set in $\R^n$ with $d<n-1$, that is, when the codimension is strictly bigger than 1. Our measure is associated to a degenerate elliptic PDE, it gives rise to a…

Random diffeomorphisms with bounded absolutely continuous noise are known to possess a finite number of stationary measures. We discuss dependence of stationary measures on an auxiliary parameter, thus describing bifurcations of families of…

Let $\psi:\mathbb R_+\to\mathbb R_+$ be a non-increasing function. A real number $x$ is said to be $\psi$-Dirichlet improvable if the system $$|qx-p|< \, \psi(t) \ \ {\text{and}} \ \ |q|<t$$ has a non-trivial integer solution for all large…

For a functional $\E$ and a peak selection that picks up a global maximum of $\E$ on varying cones, we study the convergence up to a subsequence to a critical point of the sequence generated by a mountain pass type algorithm. Moreover, by…

We provide an algorithm to approximate a finitely supported discrete measure $\mu$ by a measure $\nu_{N}$ corresponding to a set of $N$ points so that the total variation between $\mu$ and $\nu_N$ has an upper bound. As a consequence if…

Let \(Q \subseteq \mathbb{N}\) be a subset, and let \(\psi\colon \mathbb{N} \to [0, \tfrac{1}{2})\), \(\theta\colon \mathbb{N} \to \mathbb{R}\) be functions. Let \(\{A_q\}\) and \(\{B_q\}\) be sequences of integers such that \(\gcd(A_q,…

In recent work, Harman and Snowden introduced a notion of measure on a Fra\"iss\'e class $\mathfrak{F}$, and showed how such measures lead to interesting tensor categories. Constructing and classifying measures is a difficult problem, and…

This note formulates a conjecture generalizing both the abc conjecture of Masser-Oesterl\'e and the author's diophantine conjecture for algebraic points of bounded degree. It also shows that the new conjecture is implied by the earlier…