70 research outputs found
A lower bound for the size of a Minkowski sum of dilates
Let A be a finite non-empty set of integers. An asymptotic estimate of the size of the sum of several dilates was obtained by Bukh. The unique known exact bound concerns the sum |A + k·A|, where k is a prime and |A| is large. In its full generality, this bound is due to Cilleruelo, Serra and the first author.
Let k be an odd prime and assume that |A| > 8kk. A corollary to our main result states that |2·A + k·A|=(k+2)|A|-k2-k+2. Notice that |2·P+k·P|=(k+2)|P|-2k, if P is an arithmetic progression.Postprint (author's final draft
Lattice polytopes in coding theory
In this paper we discuss combinatorial questions about lattice polytopes
motivated by recent results on minimum distance estimation for toric codes. We
also prove a new inductive bound for the minimum distance of generalized toric
codes. As an application, we give new formulas for the minimum distance of
generalized toric codes for special lattice point configurations.Comment: 11 pages, 3 figure
Global residues for sparse polynomial systems
We consider families of sparse Laurent polynomials f_1,...,f_n with a finite
set of common zeroes Z_f in the complex algebraic n-torus. The global residue
assigns to every Laurent polynomial g the sum of its Grothendieck residues over
the set Z_f. We present a new symbolic algorithm for computing the global
residue as a rational function of the coefficients of the f_i when the Newton
polytopes of the f_i are full-dimensional. Our results have consequences in
sparse polynomial interpolation and lattice point enumeration in Minkowski sums
of polytopes.Comment: Typos corrected, reference added, 13 pages, 5 figures. To appear in
JPA
On distance measures for well-distributed sets
In this paper we investigate the Erd\"os/Falconer distance conjecture for a
natural class of sets statistically, though not necessarily arithmetically,
similar to a lattice. We prove a good upper bound for spherical means that have
been classically used to study this problem. We conjecture that a majorant for
the spherical means suffices to prove the distance conjecture(s) in this
setting. For a class of non-Euclidean distances, we show that this generally
cannot be achieved, at least in dimension two, by considering integer point
distributions on convex curves and surfaces. In higher dimensions, we link this
problem to the question about the existence of smooth well-curved hypersurfaces
that support many integer points
- …