194 research outputs found
The structure and number of Erd\H{o}s covering systems
Introduced by Erd\H{o}s in 1950, a covering system of the integers is a
finite collection of arithmetic progressions whose union is the set
. Many beautiful questions and conjectures about covering systems
have been posed over the past several decades, but until recently little was
known about their properties. Most famously, the so-called minimum modulus
problem of Erd\H{o}s was resolved in 2015 by Hough, who proved that in every
covering system with distinct moduli, the minimum modulus is at most .
In this paper we answer another question of Erd\H{o}s, asked in 1952, on the
number of minimal covering systems. More precisely, we show that the number of
minimal covering systems with exactly elements is as , where En route to this counting result, we obtain a
structural description of all covering systems that are close to optimal in an
appropriate sense.Comment: 31 page
Primitive geodesic lengths and (almost) arithmetic progressions
In this article, we investigate when the set of primitive geodesic lengths on
a Riemannian manifold have arbitrarily long arithmetic progressions. We prove
that in the space of negatively curved metrics, a metric having such arithmetic
progressions is quite rare. We introduce almost arithmetic progressions, a
coarsification of arithmetic progressions, and prove that every negatively
curved, closed Riemannian manifold has arbitrarily long almost arithmetic
progressions in its primitive length spectrum. Concerning genuine arithmetic
progressions, we prove that every non-compact, locally symmetric, arithmetic
manifold has arbitrarily long arithmetic progressions in its primitive length
spectrum. We end with a conjectural characterization of arithmeticity in terms
of arithmetic progressions in the primitive length spectrum. We also suggest an
approach to a well known spectral rigidity problem based on the scarcity of
manifolds with arithmetic progressions.Comment: v3: 23 pages. To appear in Publ. Ma
On covers of abelian groups by cosets
Let G be any abelian group and {a_sG_s}_{s=1}^k be a finite system of cosets
of subgroups G_1,...,G_k. We show that if {a_sG_s}_{s=1}^k covers all the
elements of G at least m times with the coset a_tG_t irredundant then
[G:G_t]\le 2^{k-m} and furthermore k\ge m+f([G:G_t]), where f(\prod_{i=1}^r
p_i^{alpha_i})=\sum_{i=1}^r alpha_i(p_i-1) if p_1,...,p_r are distinct primes
and alpha_1,...,alpha_r are nonnegative integers. This extends Mycielski's
conjecture in a new way and implies a conjecture of Gao and Geroldinger. Our
new method involves algebraic number theory and characters of abelian groups.Comment: 10 pages, also related to Number Theory and Combinatoric
Zero-sum problems for abelian p-groups and covers of the integers by residue classes
Zero-sum problems for abelian groups and covers of the integers by residue
classes, are two different active topics initiated by P. Erdos more than 40
years ago and investigated by many researchers separately since then. In an
earlier announcement [Electron. Res. Announc. Amer. Math. Soc. 9(2003), 51-60],
the author claimed some surprising connections among these seemingly unrelated
fascinating areas. In this paper we establish further connections between
zero-sum problems for abelian p-groups and covers of the integers. For example,
we extend the famous Erdos-Ginzburg-Ziv theorem in the following way: If
{a_s(mod n_s)}_{s=1}^k covers each integer either exactly 2q-1 times or exactly
2q times where q is a prime power, then for any c_1,...,c_k in Z/qZ there
exists a subset I of {1,...,k} such that sum_{s in I}1/n_s=q and sum_{s in
I}c_s=0. Our main theorem in this paper unifies many results in the two realms
and also implies an extension of the Alon-Friedland-Kalai result on regular
subgraphs
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