27 research outputs found

    Arithmetic-Progression-Weighted Subsequence Sums

    Full text link
    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

    Large restricted sumsets in general abelian group

    Get PDF
    Let A, B and S be three subsets of a finite Abelian group G. The restricted sumset of A and B with respect to S is defined as A\wedge^{S} B= {a+b: a in A, b in B and a-b not in S}. Let L_S=max_{z in G}| {(x,y): x,y in G, x+y=z and x-y in S}|. A simple application of the pigeonhole principle shows that |A|+|B|>|G|+L_S implies A\wedge^S B=G. We then prove that if |A|+|B|=|G|+L_S then |A\wedge^S B|>= |G|-2|S|. We also characterize the triples of sets (A,B,S) such that |A|+|B|=|G|+L_S and |A\wedge^S B|= |G|-2|S|. Moreover, in this case, we also provide the structure of the set G\setminus (A\wedge^S B).Comment: Paper submitted November 15, 2011. To appear European Journal of Combinatorics, special issue in memorian Yahya ould Hamidoune (2013

    Artin's primitive root conjecture -a survey -

    Get PDF
    This is an expanded version of a write-up of a talk given in the fall of 2000 in Oberwolfach. A large part of it is intended to be understandable by non-number theorists with a mathematical background. The talk covered some of the history, results and ideas connected with Artin's celebrated primitive root conjecture dating from 1927. In the update several new results established after 2000 are also discussed.Comment: 87 pages, 512 references, to appear in Integer
    corecore