Related papers: Roth-type theorems in additive combinatroics
We introduce a wide class of deterministic subsets of primes of zero relative density and we prove Roth's Theorem in these sets, namely, we show that any subset of them with positive relative upper density contains infinitely many…
A simple formal recasting of well known arguments concerning the ordering problems of General Relativity allows to obtain in such a context a Gr\"{o}enewald Van Hove like theorem.
Bott periodicity plays an important role in topological K-theory. The purpose of this paper is to extend the periodicity theorem in a discrete context, where all classical groups are involved and not just the general linear group. The…
In 1975 Szemer\'edi proved the long-standing conjecture of Erd\H{o}s and Tur\'an that any subset of $\bbZ$ having positive upper Banach density contains arbitrarily long arithmetic progressions. Szemer\'edi's proof was entirely…
In this paper, we clarify the mistakes made in the former article entitled "Vanishing Theorems on Toric Varieties in Positive Characteristic". On the other hand, we use the positive characteristic method to reprove the Bott vanishing…
Let $p$ be an odd prime. From a simple undirected graph $G$, through the classical procedures of Baer (Trans. Am. Math. Soc., 1938), Tutte (J. Lond. Math. Soc., 1947) and Lov\'asz (B. Braz. Math. Soc., 1989), there is a $p$-group $P_G$ of…
The characterization of commutators in associative algebras is a classical problem in ring theory. In this paper, we address this problem for the natural class of generalized block-triangular algebras. To this end, we introduce a new…
We investigate the combinatorial sequences $A(M, n)$ introduced by W. G. Brown (1964) and W. T. Tutte (1980) appearing in enumeration of convex polyhedra. Their formula is $$A(M, n) = \frac{2 (2M+3)!}{(M+2)! M!}\,\frac{(4n+2M+1)!}{n! (3n +…
The "unit theorem" to which the present mini-course is devoted is a theorem from algebra that has a combinatorial flavour, and that originated in fact from algebraic combinatorics. Beyond a proof, the course also addresses applications, one…
A determinant evaluation is proven, a special case of which establishes a conjecture of Bombieri, Hunt, and van der Poorten (Experimental Math\. {\bf 4} (1995), 87--96) that arose in the study of Thue's method of approximating algebraic…
Hindman's Theorem is a prototypical example of a combinatorial theorem with a proof that uses the topology of the ultrafilters. We show how the methods of this proof, including topological arguments about ultrafilters, can be translated…
The class of LOTS (linearly ordered topological spaces, i.e. spaces equipped with a topology generated by a linear order) contains many important spaces, like the set of real numbers, the set of rational numbers and the ordinals. Such…
We define a new congruence relation on the set of integers, leading to a group similar to the multiplicative group of integers modulo $n$. It makes use of a symmetry almost omnipresent in modular multiplications and halves the number of…
In the present article we consider the uniqueness problem for the generalized Radon transform arising in a mathematical model of production. We prove uniqueness theorems for this transform and for the profit function in the corresponding…
We develop a Van der Waerden type theorem in an axiomatic setting of graded lattices and show that this axiomatic formulation can be applied to various lattices, for instance the set partition and the Boolean lattices. We derive the…
This paper solves the first of the open problems in topos theory posted by William Lawvere, concerning the existence of a Grothendieck topos that has proper class many quotient topoi. This paper concretely constructs such Grothendieck…
A new tool is introduced, specially tailored for the study of order 2 bases of N. It is called the Erdos-Tuan algebra since it suits well their conjecture.
Rota's basis conjecture, open since 1989, states that if B_1, B_2, ..., B_n are n bases of a vector space of rank n, then there is an nxn grid of vectors such that the vectors in the ith row are precisely the elements of B_i and such that…
In the 1950s and 1960s Tate proved some duality theorems in the Galois cohomology of finite modules and abelian varieties. As for most of Tate's work this has had a profound influence on mathematics with many applications and further…
We combine the language of monoids with the language of preorders so as to refine some fundamental aspects of the classical theory of factorization and prove an abstract factorization theorem with a variety of applications. In particular,…