Related papers: VC-Density in Divisible Oriented Abelian Groups an…

We give a new proof of VC bounds where we avoid the use of symmetrization and use a shadow sample of arbitrary size. We also improve on the variance term. This results in better constants, as shown on numerical examples. Moreover our bounds…

The VC-dimension of a family of sets is a measure of its combinatorial complexity used in machine learning theory, computational geometry, and even model theory. Computing the VC-dimension of the $k$-fold union of geometric set systems has…

An almost Abelian Lie group is a non-Abelian Lie group with a codimension 1 Abelian subgroup. We show that all discrete subgroups of complex simply connected almost Abelian groups are finitely generated. The topology of connected almost…

We prove that an isomorphism between saturated fusion systems over the same finite p-group is detected on the elementary abelian subgroups of the hyperfocal subgroup if p is odd, and on the abelian subgroups of the hyperfocal subgroup of…

Let $H$ be an abelian subgroup of a finite group $G$ and $\pi$ the set of prime divisors of $|H|$. We prove that $|H O_{\pi}(G)/ O_{\pi}(G)|$ is bounded above by the largest character degree of $G$. A similar result is obtained when $H$ is…

A collection of varieties satisfies uniform potential density if each of them contains a dense subset of rational points after extending its ground field by a bounded degree. In this paper, we prove that uniform potential density holds for…

We study the Vapnik-Chervonenkis (VC) dimension of the set of quadratic residues (i.e. squares) in finite fields, $\mathbb F_q$, when considered as a subset of the additive group. We conjecture that as $q \to \infty$, the squares have the…

We prove that, in a finitely generated residually finite group of subexponential growth, the proportion of commuting pairs is positive if and only if the group is virtually abelian. In particular, this covers the case where the group has…

Consider an expansion $\mathcal R=(R,<,+,\ldots)$ of an ordered divisible Abelian group of finite burden defining no nonempty subset $X$ of $R$ which is dense and codense in a definable open subset $U$ of $R$ with $X \subseteq U$. We…

We give a density condition for when, subject to a necessary parity condition, an eulerian graph or digraph may be cellularly embedded in an orientable surface so that it has exactly two faces, each bounded by an euler circuit, one of which…

We give a geometric description of the pair $(V,p)$, where $V$ is an affine algebraic variety over a non-trivially valued algebraically closed field $K$ with valuation ring $\mathcal{O}_K$ and $p$ is a Zariski dense generically stable type…

Let G be a group and V a finite dimensional representation of G over an algebraically closed field k of characteristic p>0. Let $d_n(V)$ be the number of indecomposable summands of $V^{\otimes n}$ of nonzero dimension mod p. It is easy to…

In this paper, we initiate the systematic study of density of algebraic points on surfaces. We give an effective asymptotic range in which the density degree set has regular behavior dictated by the index. By contrast, in small degree, the…

Given $A\subseteq \mathbb{Z}$, the ratio set or the quotient set of $A$ is defined by $R(A):=\{a/b: a, b\in A, b\neq 0\}$. It is an open problem to study the denseness of $R(A)$ in the $p$-adic numbers when $A$ is the set of values attained…

Let $A$ be a subset of a finite abelian group such that $A$ has a small difference set $A-A$ and the density of $A$ is small. We prove that, counter--intuitively, the smallness (in terms of $|A-A|$) of the Fourier coefficients of $A$…

We give bounds on Kazhdan constants of abelian extensions of (finite) groups. As a corollary, we improved known results of Kazhdan constants for some meta-abelian groups and for the relatively free group in the variety of $p$-groups of…

This paper discusses sufficient conditions for a definably complete densely linearly ordered expansion of an abelian group having the uniformly locally o-minimal open cores of the first/second kind and strongly locally o-minimal open core,…

Twin-width is a recently introduced graph parameter with applications in algorithmics, combinatorics, and finite model theory. For graphs of bounded degree, finiteness of twin-width is preserved by quasi-isometry. Thus, through Cayley…

We determine all restrictions on the dimension of the fixed locus of a diagonalizable group acting on a smooth projective variety that arise from the Chern numbers of the ambient variety. We reduce the problem to finding lower bounds for…

We present various results about Euclidean preferences in the plane under $\ell_1$, $\ell_2$ and $\ell_{\infty}$ norms. When there are four candidates, we show that the maximal size (in terms of the number of pairwise distinct preferences)…