Variations on the Erdős distinct-sums problem

Abstract

Let {a 1 , ..., an} be a set of positive integers with a 1 < • • • < an such that all 2 n subset sums are distinct. A famous conjecture by Erdős states that an > c • 2 n for some constant c, while the best result known to date is of the form an > c • 2 n / √ n. In this paper, we weaken the condition by requiring that only sums corresponding to subsets of size smaller than or equal to λn be distinct. For this case, we derive lower and upper bounds on the smallest possible value of an.

On Algebraic Constructions of Neural Networks with Small Weights

“…It is shown that if m = o(n/ log nW ), then any A ∈ Z m×n with weight constraint W admits a non-trivial solution in {−1, 0, 1} n and cannot be an EQ matrix, i.e., R = O(log nW ) is tight [16]. A similar result can be obtained by the matrix generalizations of Erdős' Distinct Subset Sum problem [6].…”

“…As the weights of w b define a bijection between n-bit binary vectors and integers, w T b x = 0 does not admit a non-trivial solution for the alphabet {−1, 0, 1} n . This is a necessary and sufficient condition to compute the EQ function given in (6). We extend this property to m many rows to define EQ matrices which give a bijection between {0, 1} n and Z m .…”

“…We extend this property to m many rows to define EQ matrices which give a bijection between {0, 1} n and Z m . Thus, an EQ matrix can be used to compute the EQ function in (6) and our goal is to use smaller weight sizes in the matrix.…”