157,424 research outputs found
On covering by translates of a set
In this paper we study the minimal number of translates of an arbitrary
subset of a group needed to cover the group, and related notions of the
efficiency of such coverings. We focus mainly on finite subsets in discrete
groups, reviewing the classical results in this area, and generalizing them to
a much broader context. For example, we show that while the worst-case
efficiency when has elements is of order , for fixed and
large, almost every -subset of any given -element group covers
with close to optimal efficiency.Comment: 41 pages; minor corrections; to appear in Random Structures and
Algorithm
On some covering problems in geometry
We present a method to obtain upper bounds on covering numbers. As
applications of this method, we reprove and generalize results of Rogers on
economically covering Euclidean -space with translates of a convex body, or
more generally, any measurable set. We obtain a bound for the density of
covering the -sphere by rotated copies of a spherically convex set (or, any
measurable set). Using the same method, we sharpen an estimate by
Artstein--Avidan and Slomka on covering a bounded set by translates of another.
The main novelty of our method is that it is not probabilistic. The key idea,
which makes our proofs rather simple and uniform through different settings, is
an algorithmic result of Lov\'asz and Stein.Comment: 9 pages. IMPORTANT CHANGE: In previous versions of the paper, the
illumination problem was also considered, and I presented a construction of a
body close to the Euclidean ball with high illumination number. Now, I
removed this part from this manuscript and made it a separate paper, 'A Spiky
Ball'. It can be found at http://arxiv.org/abs/1510.0078
Locality in Sumsets
Motivated by the Polynomial Freiman-Ruzsa (PFR) Conjecture, we develop a
theory of locality in sumsets, with applications to John-type approximation and
sets with small doubling. First we show that if with
is non-degenerate then is covered by
translates of a -dimensional generalised arithmetic progression
(-GAP) with ; thus we obtain one of the
polynomial bounds required by PFR, under the non-degeneracy assumption that
is not efficiently covered by translates of a -GAP.
We also prove a stability result showing for any that if with is non-degenerate then
some with is efficiently covered by either
a -GAP or translates of a -GAP. This `dimension-free'
bound for approximate covering makes for a stark contrast with exact covering,
where the required number of translates grows exponentially with . We
further show that if is non-degenerate with and then is covered by
translates of a -GAP with ; this is tight, in that
cannot be replaced by any smaller number. The above results also hold
for , replacing GAPs by a suitable common
generalisation of GAPs and convex bodies. In this setting the non-degeneracy
condition holds automatically, so we obtain essentially optimal bounds with no
additional assumption on . These results are all deduced from a unifying
theory, in which we introduce a new intrinsic structural approximation of any
set, which we call the `additive hull', and develop its theory via a refinement
of Freiman's theorem with additional separation properties.Comment: 36 pages, comments welcom
Competition, wages and teacher sorting: four lessons learned from a voucher reform
This paper studies how local school competition affects teacher wages at markets where wages are set via individual wage bargaining. Using regional variation in private school entry generated by a Swedish reform which allowed private schools to enter freely and a comprehensive matched employer employee data covering all high school teachers in Sweden over 16 years, I analyze the effects of competition on wages as well as labor flows. The results suggest that competition translates into higher wages, also for teachers in public schools. While the average increases are modest new teachers gain 2 percent and high ability teachers in math and science receive 4 percent higher wages in the most competitive areas compared to areas without any competition from private schools. Several robustness checks support a causal interpretation of the results which together highlight the potential gains from school competition through a more differentiated wage setting of teachers.Private school competition; teacher wages; monopsony power
Unsplittable coverings in the plane
A system of sets forms an {\em -fold covering} of a set if every point
of belongs to at least of its members. A -fold covering is called a
{\em covering}. The problem of splitting multiple coverings into several
coverings was motivated by classical density estimates for {\em sphere
packings} as well as by the {\em planar sensor cover problem}. It has been the
prevailing conjecture for 35 years (settled in many special cases) that for
every plane convex body , there exists a constant such that every
-fold covering of the plane with translates of splits into
coverings. In the present paper, it is proved that this conjecture is false for
the unit disk. The proof can be generalized to construct, for every , an
unsplittable -fold covering of the plane with translates of any open convex
body which has a smooth boundary with everywhere {\em positive curvature}.
Somewhat surprisingly, {\em unbounded} open convex sets do not misbehave,
they satisfy the conjecture: every -fold covering of any region of the plane
by translates of such a set splits into two coverings. To establish this
result, we prove a general coloring theorem for hypergraphs of a special type:
{\em shift-chains}. We also show that there is a constant such that, for
any positive integer , every -fold covering of a region with unit disks
splits into two coverings, provided that every point is covered by {\em at
most} sets
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