On the minimum size of subset and subsequence sums in integers
Jagannath Bhanja, Ram Krishna Pandey · Comptes Rendus Mathématique · 2022
Let 𝒜 be a sequence of r k terms which is made up of k distinct integers each appearing exactly r times in 𝒜 . The sum of all terms of a subsequence of 𝒜 is called a subsequence sum of 𝒜 . For a nonnegative integer α ≤ r k , let Σ α ( 𝒜 ) be the set of all subsequence sums of 𝒜 that correspond to the subsequences of length α or more. When r = 1 , we call the subsequence sums as subset sums and we write Σ α ( A ) for Σ α ( 𝒜 ) . In this article, using some simple combinatorial arguments, we establish optimal lower bounds for the size of Σ α ( A ) and Σ α ( 𝒜 ) . As special cases, we also obtain some already known results in this study.