27 research outputs found
Arithmetic-Progression-Weighted Subsequence Sums
Let be an abelian group, let be a sequence of terms
not all contained in a coset of a proper subgroup of
, and let be a sequence of consecutive integers. Let
which is a particular kind of weighted restricted sumset. We show that , that if , and also
characterize all sequences of length with . This
result then allows us to characterize when a linear equation
where are
given, has a solution modulo with all
distinct modulo . As a second simple corollary, we also show that there are
maximal length minimal zero-sum sequences over a rank 2 finite abelian group
(where and ) having
distinct terms, for any . 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
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 -
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