On sums of distinct representatives

On Sums of Distinct Representatives

Clearly (1) has an SDR provided that |Ai| > i for all i = 1, · · · , n, in particular an SDR of (1) exists if |A1| = · · · = |An| > n or 0 < |A1| < · · · < |An|. Let G be an additive abelian group and A1, · · · , An its subsets. We associate any SDR (2) of (1) with the sum ∑n i=1 ai and set (4) S({Ai}i=1) = S(A1, · · · , An) = {a1 + · · ·+ an : {ai}i=1 forms an SDR of {Ai}i=1} . Of course, S(A1...

on direct sums of baer modules

the notion of baer modules was defined recently

On distinct sums and distinct distances

The paper (Discrete Comput. Geom. 25 (2001) 629) of Solymosi and Tóth implicitly raised the following arithmetic problem. Consider n pairwise disjoint s element sets and form all ð 2Þn sums of pairs of elements of the same set. What is the minimum number of distinct sums one can get this way? This paper proves that the number of distinct sums is at least ns ; where ds 1⁄4 1=cJs=2n is defined in...

On sums and products of distinct numbers

Let A be a set of k complex numbers, and let A (respectively, A×) be the set of sums (resp. products) of distinct elements of A. Let gC(k) = min A⊂C,|A|=k {|A+|+ |A×|}. Ruzsa posed the question whether gC(k) grows faster than any power of k. In this note we give an affirmative answer to this question. Let A be a set of k complex numbers, and let A and A× be the sets of sums and products of dist...

Sums of distinct squares