In this note, we show that the method of Croot, Lev, and Pach can be used to bound the size of a subset of F n q with no three terms in arithmetic progression by c n with c < q. For q = 3, the problem of finding the largest subset of F n 3 with no three terms in arithmetic progression is called the cap problem. Previously the best known upper bound for the affine cap problem, due to Bateman and Katz [BK12], was on order n −1−ǫ 3 n .The problem of finding large subsets of an abelian group G with no three-term arithmetic progression, or of finding upper bounds for the size of such a subset, has a long history in number theory. The most intense attention has centered on the cases where G is a cyclic group Z/NZ or a vector space (Z/3Z) n , which are in some sense the extreme situations. We denote by r 3 (G) the maximal size of a subset of G with no three-term arithmetic progression. The fact that r) was first proved by Brown and Buhler [BB82], which was improved to O(3 n /n) by Meshulam [Mes95]. The best known upper bound, O(3 n /n 1+ǫ ), is due to Bateman and Katz [BK12]. The best lower bound, by contrast, is around 2.2 n [Ede04].The problem of arithmetic progressions in (Z/3Z) n has sometimes been seen as a model for the corresponding problem in Z/NZ. We know (for instance, by a construction of Behrend [Beh46]) that r 3 (Z/NZ) grows more quickly than N 1−ǫ for every ǫ > 0. Thus it is natural to ask whether r 3 ((Z/3Z) n ) grows more quickly than (3 − ǫ) n for every ǫ > 0. In general, there has been no consensus on what the answer to this question should be.In the present paper we settle the question, proving that for all odd primes p, r 3 ((Z/pZ) n ) 1/n is bounded away from p as n grows.The main tool used here is the polynomial method, in particular the use of the polynomial method developed in the breakthrough paper of Croot, Lev, and Pach [CLP16], which drastically improved the best known upper bounds for r 3 ((Z/4Z) n ). In this case, they show that a subset of G with no three-term arithmetic progression has size at most c n for some c < 4. In the present paper, we show that the ideas of their paper can be extended to vector spaces over a general finite field.
Summary. This chapter provides the reader with the necessary background for dealing with semidefinite programs which have symmetry. The basic theory is given and it is illustrated in applications from coding theory, combinatorics, geometry, and polynomial optimization.
We study the capacitated k-facility location problem, in which we are given a set of clients with demands, a set of facilities with capacities and a constant number k. It costs f i to open facility i, and c ij for facility i to serve one unit of demand from client j. The objective is to open at most k facilities serving all the demands and satisfying the capacity constraints while minimizing the sum of service and opening costs.In this paper, we give the first fully polynomial time approximation scheme (FPTAS) for the single-sink (single-client) capacitated k-facility location problem. Then, we show that the capacitated k-facility location problem with uniform capacities is solvable in polynomial time if the number of clients is fixed by reducing it to a collection of transportation problems. Third, we analyze the structure of extreme point solutions, and examine the efficiency of this structure in designing approximation algorithms for capacitated k-facility location problems. Finally, we extend our results to obtain an improved approximation algorithm for the capacitated facility location problem with uniform opening cost.
We give a new upper bound on the maximum size A q (n, d) of a code of word length n and minimum Hamming distance at least d over the alphabet of q 3 letters. By block-diagonalizing the Terwilliger algebra of the nonbinary Hamming scheme, the bound can be calculated in time polynomial in n using semidefinite programming. For q = 3, 4, 5 this gives several improved upper bounds for concrete values of n and d. This work builds upon previous results of Schrijver [A. Schrijver, New code upper bounds from the Terwilliger algebra and semidefinite programming, IEEE Trans. Inform. Theory 51 (2005) 2859-2866] on the Terwilliger algebra of the binary Hamming scheme.
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