7,297 research outputs found

    Classification theorems for sumsets modulo a prime

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    Let Z/pZ\Z/pZ be the finite field of prime order pp and AA be a subsequence of Z/pZ\Z/pZ. We prove several classification results about the following questions: (1) When can one represent zero as a sum of some elements of AA ? (2) When can one represent every element of Z/pZ\Z/pZ as a sum of some elements of AA ? (3) When can one represent every element of Z/pZ\Z/pZ as a sum of ll elements of AA ?Comment: 35 pages, to appear in JCT

    Arithmetic-Progression-Weighted Subsequence Sums

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    Let GG be an abelian group, let SS be a sequence of terms s1,s2,...,snGs_1,s_2,...,s_{n}\in G not all contained in a coset of a proper subgroup of GG, and let WW be a sequence of nn consecutive integers. Let WS={w1s1+...+wnsn:  wiatermofW,wiwjforij},W\odot S=\{w_1s_1+...+w_ns_n:\;w_i {a term of} W,\, w_i\neq w_j{for} i\neq j\}, which is a particular kind of weighted restricted sumset. We show that WSmin{G1,n}|W\odot S|\geq \min\{|G|-1,\,n\}, that WS=GW\odot S=G if nG+1n\geq |G|+1, and also characterize all sequences SS of length G|G| with WSGW\odot S\neq G. This result then allows us to characterize when a linear equation a1x1+...+arxrαmodn,a_1x_1+...+a_rx_r\equiv \alpha\mod n, where α,a1,...,arZ\alpha,a_1,..., a_r\in \Z are given, has a solution (x1,...,xr)Zr(x_1,...,x_r)\in \Z^r modulo nn with all xix_i distinct modulo nn. As a second simple corollary, we also show that there are maximal length minimal zero-sum sequences over a rank 2 finite abelian group GCn1Cn2G\cong C_{n_1}\oplus C_{n_2} (where n1n2n_1\mid n_2 and n23n_2\geq 3) having kk distinct terms, for any k[3,min{n1+1,exp(G)}]k\in [3,\min\{n_1+1,\,\exp(G)\}]. Indeed, apart from a few simple restrictions, any pattern of multiplicities is realizable for such a maximal length minimal zero-sum sequence

    On n-sum of an abelian group of order n

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    Let GG be an additive finite abelian group of order nn, and let SS be a sequence of n+kn+k elements in GG, where k1k\geq 1. Suppose that SS contains tt distinct elements. Let n(S)\sum_n(S) denote the set that consists of all elements in GG which can be expressed as the sum over a subsequence of length nn. In this paper we prove that, either 0n(S)0\in \sum_n(S) or n(S)k+t1.|\sum_n(S)|\geq k+t-1. This confirms a conjecture by Y.O. Hamidoune in 2000

    Representation of Finite Abelian Group Elements by Subsequence Sums

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    Let GCn1...CnrG\cong C_{n_1}\oplus ... \oplus C_{n_r} be a finite and nontrivial abelian group with n1n2...nrn_1|n_2|...|n_r. A conjecture of Hamidoune says that if W=w1...wnW=w_1... w_n is a sequence of integers, all but at most one relatively prime to G|G|, and SS is a sequence over GG with SW+G1G+1|S|\geq |W|+|G|-1\geq |G|+1, the maximum multiplicity of SS at most W|W|, and σ(W)0modG\sigma(W)\equiv 0\mod |G|, then there exists a nontrivial subgroup HH such that every element gHg\in H can be represented as a weighted subsequence sum of the form g=i=1nwisig=\sum_{i=1}^{n}w_is_i, with s1...sns_1... s_n a subsequence of SS. We give two examples showing this does not hold in general, and characterize the counterexamples for large W1/2G|W|\geq {1/2}|G|. A theorem of Gao, generalizing an older result of Olson, says that if GG is a finite abelian group, and SS is a sequence over GG with SG+D(G)1|S|\geq |G|+D(G)-1, then either every element of GG can be represented as a G|G|-term subsequence sum from SS, or there exists a coset g+Hg+H such that all but at most G/H2|G/H|-2 terms of SS are from g+Hg+H. We establish some very special cases in a weighted analog of this theorem conjectured by Ordaz and Quiroz, and some partial conclusions in the remaining cases, which imply a recent result of Ordaz and Quiroz. This is done, in part, by extending a weighted setpartition theorem of Grynkiewicz, which we then use to also improve the previously mentioned result of Gao by showing that the hypothesis SG+D(G)1|S|\geq |G|+D(G)-1 can be relaxed to SG+d(G)|S|\geq |G|+d^*(G), where d^*(G)=\Sum_{i=1}^{r}(n_i-1). We also use this method to derive a variation on Hamidoune's conjecture valid when at least d(G)d^*(G) of the wiw_i are relatively prime to G|G|
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