A quadratic lower bound for subset sums
نویسندگان
چکیده
Let A be a finite nonempty subset of an additive abelian group G, and let Σ(A) denote the set of all group elements representable as a sum of some subset of A. We prove that |Σ(A)| ≥ |H|+ 1 64 |A \H|2 where H is the stabilizer of Σ(A). Our result implies that Σ(A) = Z/nZ for every set A of units of Z/nZ with |A| ≥ 8√n. This consequence was first proved by Erdős and Heilbronn for n prime, and by Vu (with a weaker constant) for general n.
منابع مشابه
A New Lower Bound Via Projection for the Quadratic Assignment Problem
New lower bounds for the quadratic assignment problem QAP are presented. These bounds are based on the orthogonal relaxation of QAP. The additional improvement is obtained by making eecient use of a tractable representation of orthogonal matrices having constant row and column sums. The new bound is easy to implement and often provides high quality bounds under an acceptable computational eeort.
متن کاملSubset Sums Avoiding Quadratic Nonresidues
It is a well-known problem to give an estimate for the largest clique of the Paley-graph, i.e. , to give an estimate for |A| if A ⊂ Fp (p ≡ 1 (mod 4)) is such that A−A = {a−a′ |a, a′ ∈ A} avoids the set of quadratic nonresidues. In this paper we will study a much simpler problem namely when A− A is substituted by the set FS(A) = { ∑ εaa | εa = 0 or 1 and ∑ εa > 0}. In other words we will estima...
متن کاملAsymptotically tight bounds on subset sums
For a subset A of a finite abelian group G we define Σ(A) = {∑a∈B a : B ⊂ A}. In the case that Σ(A) has trivial stabiliser, one may deduce that the size of Σ(A) is at least quadratic in |A|; the bound |Σ(A)| ≥ |A|2/64 has recently been obtained by De Vos, Goddyn, Mohar and Šámal [2]. We improve this bound to the asymptotically best possible result |Σ(A)| ≥ (1/4− o(1))|A|2. We also study a relat...
متن کاملModular Invariants for Real Quadratic Fields and Kloosterman Sums
We investigate the asymptotic distribution of integrals of the j-function that are associated to ideal classes in a real quadratic field. To estimate the error term in our asymptotic formula, we prove a bound for sums of Kloosterman sums of half-integral weight which is uniform in every parameter. To establish this estimate we prove a variant of Kuznetsov’s formula where the spectral data is re...
متن کاملMaximal Independent Sets for the Pixel Expansion of Graph Access Structure
Abstract : A visual cryptography scheme based on a given graph G is a method to distribute a secret image among the vertices of G, the participants, so that a subset of participants can recover the secret image if they contain an edge of G, by stacking their shares, otherwise they can obtain no information regarding the secret image. In this paper a maximal independent sets of the graph G was ...
متن کامل