Optimal Representations by Sumsets and Subset Sums Original Research Article

LetAbe a set of non-negative integers. In our previous paper forhset membership, variantimage,hgreater-or-equal, slanted2 we estimated from below the cardinality ofhAand this allowed us to obtain sharp estimates for suchG, that every integerggreater-or-equal, slantedGmay be represented by a sum of elements ofA(the linear diophantine problem of Frobenius). However, “not only the number of elements ofhAis important, but also the way they are situated” (G. Freiman). In this paper we show thathAalways contains long continuous chains of integers and derive an estimation for the number of summands required for representation ofgof the above mentioned type.