Related papers: Note on the Additive Basis Conjecture

We prove an analogue of the Oppenheim conjecture for a system comprising an inhomogeneous quadratic form and a linear form in $3$ variables using dynamics on the space of affine lattices.

Suppose that a binary operation $\circ$ on a finite set $X$ is injective in each variable separately and also associative. It is easy to prove that $(X,\circ)$ must be a group. In this paper we examine what happens if one knows only that a…

Linear logic is a substructural logic proposed as a refinement of classical and intuitionistic logics, with applications in programming languages, game semantics, and quantum physics. We present a template for Gentzen-style linear logic…

In this paper we prove a conjecture about the dimension of linear systems of surfaces of degree d in P^3 through at most eight multiple points in general position.

We provide a lower bound construction showing that the union of unit balls in three-dimensional space has quadratic complexity, even if they all contain the origin. This settles a conjecture of Sharir.

Given a finite set of roots of unity, we show that all power sums are non-negative integers iff the set forms a group under multiplication. The main argument is purely combinatorial and states that for an arbitrary finite set system the…

There are many major open problems in integer flow theory, such as Tutte's 3-flow conjecture that every 4-edge-connected graph admits a nowhere-zero 3-flow, Jaeger et al.'s conjecture that every 5-edge-connected graph is $Z_3$-connected and…

We study a generalization of additive bases into a planar setting. A planar additive basis is a set of non-negative integer pairs whose vector sumset covers a given rectangle. Such bases find applications in active sensor arrays used in,…

Given a block of a finite group, any source algebra has a basis invariant under the multiplicative actions of the defect group. Is such a basis a characteristic biset of the block fusion system? If the basis can be chosen to consist…

We prove the arithmetic fundamental lemma conjecture over a general $p$-adic field with odd residue cardinality $q\geq \dim V$. Our strategy is similar to the one used by the second author during his proof of the AFL over $\mathbb{Q}_p$…

Let $A$ be an additive basis. We call ``essential subset'' of $A$ any finite subset $P$ of $A$ such that $A \setminus P$ is not an additive basis and that $P$ is minimal (for the inclusion order) to have this property. A recent theorem due…

In this note we make use of some properties of vector fields on a manifold to give an alternate proof to [3] for the equivalence between connections and parallel transport on vector bundles over manifolds. Out of the proof will emerge a new…

Prompted by results of Guardo, Van Tuyl and the second author for lines in projective 3 space, we develop asymptotic upper bounds for the least degree of a homogeneous form vanishing to order at least m on a union of disjoint r dimensional…

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…

Motivated by a problem in additive Ramsey theory, we extend Todorcevic's partitions of three-dimensional combinatorial cubes to handle additional three-dimensional objects. As a corollary, we get that if the continuum hypothesis fails, then…

This article contains a proof of the MDS conjecture for $k \leq 2p-2$. That is, that if $S$ is a set of vectors of ${\mathbb F}_q^k$ in which every subset of $S$ of size $k$ is a basis, where $q=p^h$, $p$ is prime and $q$ is not and $k \leq…

We prove Viehweg's hyperbolicity conjecture over compact bases and over bases with non-uniruled compactification. The most general case of the conjecture states that the the base space of a maximal variation family of smooth projective…

In [10] the third author of this paper presented two conjectures on the additive decomposability of the sequence of ''smooth'' (or ''friable'') numbers. Elsholtz and Harper [4] proved (by using sieve methods) the second (less demanding)…

The aim of this article is to present a topological tool for the study of additive basis in additive number theory. It will be proposal a metric for the set of all additive basis, in which it will be possible to study properties of some…

G\'amez-Merino, Munoz-Fern\'andez and Seoane-Sep\'ulveda proved that if additivity $\mathcal A(\mathcal F)>\mathfrak c$, then $\mathcal F$ is $\mathcal A(\mathcal F)$-lineable where $\mathcal F\subseteq\mathbb R^\mathbb R$. They asked if…