Related papers: Additive bases, coset covers, and non-vanishing li…

This is the fourth in a series of articles devoted to showing that a typical covering map of large degree to a fixed, regular graph has its new adjacency eigenvalues within the bound conjectured by Alon for random regular graphs. In this…

We tackle the "relative" Lehmer problem on algebraic subvarieties of a multiplicative torus. Generalizing a theorem of F. Amoroso and U. Zannier, we give a lower bound for the normalized height of a non torsion hypersurface in terms of its…

From the generalized Riemann hypothesis for motivic L-functions, we derive an effective version of the Sato-Tate conjecture for an abelian variety A defined over a number field k with connected Sato-Tate group. By effective we mean that we…

We give a short proof that Strassen's asymptotic rank conjecture implies that for every $\varepsilon > 0$ there exists a $(3/2^{2/3} + \varepsilon)^n$-time algorithm for set cover on a universe of size $n$ with sets of bounded size. This…

We prove a common refinement of theorems of Bergfalk and of Casarosa and Lambie-Hanson, showing that under certain hypotheses, the higher derived limits of a certain inverse system of abelian groups $\mathbf{A}$ do not vanish. The refined…

Let $p$ be a prime, let $r$ and $q$ be powers of $p$, and let $a$ and $b$ be relatively prime integers not divisible by $p$. Let $C/\mathbb F_{r}(t)$ be the superelliptic curve with affine equation $y^b+x^a=t^q-t$. Let $J$ be the Jacobian…

In this note we deduce a strengthening of the Orlicz-Pettis theorem from the It\^o-Nisio theorem. The argument shows that given any series in a Banach space which isn't summable (or more generally unconditionally summable), we can construct…

In 2012 Raghavan, Samuel, and Subrahmanyam showed that the Kazhdan--Lusztig basis for the Iwahori--Hecke algebra in type A provides a ``canonical'' basis for the centraliser algebra of the Schur algebra acting on tensor space. In 2022 the…

The task of finding an extension to a given partial drawing of a graph while adhering to constraints on the representation has been extensively studied in the literature, with well-known results providing efficient algorithms for…

Let $p\geq 3$ be a prime and $n\geq 1$ be an integer. Let $K\subseteq {\mathbb{F}_p}$ denote a fixed subset with $0\in K$. Let $A\subseteq ({\mathbb{F}_p})^n$ be an arbitrary subset such that $$\{…

We introduce the theory of div point sets, which aims to provide a framework to study the combinatoric nature of any set of points in general position on an Euclidean plane. We then show that proving the unsatisfiability of some first-order…

We show that a weak version of the canonical bundle formula holds for fibrations of relative dimension one. We provide various applications thereof, for instance, using the recent result of Xu and Zhang, we prove the log non-vanishing…

Given a subset $W$ of an abelian group $G$, a subset $C$ is called an additive complement for $W$ if $W+C=G$; if, moreover, no proper subset of $C$ has this property, then we say that $C$ is a minimal complement for $W$. It is natural to…

Given a rank 3 real arrangement $\mathcal A$ of $n$ lines in the projective plane, the Dirac-Motzkin conjecture (proved by Green and Tao in 2013) states that for $n$ sufficiently large, the number of simple intersection points of $\mathcal…

Suppose $c_1,\ldots,c_{n+k}$ are real numbers, $\{a_1,\ldots,a_{n+k}\}\!\subset\!\mathbb{R}^n$ is a set of points not all lying in the same affine hyperplane, $y\!\in\!\mathbb{R}^n$, $a_j\cdot y$ denotes the standard real inner product of…

Let $G$ be a finite abelian group and $p$ be the smallest prime dividing $|G|$. Let $S$ be a sequence over $G$. We say that $S$ is regular if for every proper subgroup $H \subsetneq G$, $S$ contains at most $|H|-1$ terms from $H$. Let…

Ryser's conjecture says that for every $r$-partite hypergraph $H$ with matching number $\nu(H)$, the vertex cover number is at most $(r-1)\nu(H)$. This far reaching generalization of K\"onig's theorem is only known to be true for $r\leq 3$,…

The Addition Theorem for the algebraic entropy of group endomorphisms of torsion abelian groups was proved by Dikranjan, Goldsmith, Salce and Zanardo. It was later extended by Shlossberg to torsion nilpotent groups of class 2. As our main…

In additive number theory, a finite set $A$ of integers is an $h$-basis for $n$ if every integer in $\{0,1,2,\ldots, n\}$ can be represented as the sum of exactly $h$ not necessarily distinct elements of $A$. This paper introduces a new…

It was shown by V. Bergelson that any set B with positive upper multiplicative density contains nicely intertwined arithmetic and geometric progressions: For each positive integer k there exist integers a,b,d such that $ {b(a+id)^j:i,j…