On sumfree subsets of hypercubes

TL;DR

This paper investigates the maximum sizes of sumfree subsets within high-dimensional hypercubes, providing bounds and exploring continuous analogues and generalizations to l-fold-sumfree sets.

Contribution

It establishes new upper and lower bounds for sumfree sets in hypercubes and extends the analysis to continuous cases and l-fold-sumfree generalizations.

Findings

Derived bounds for sumfree sets in hypercubes as n grows large.

Extended results to continuous analogues of the problem.

Generalized the problem to l-fold-sumfree sets.

Abstract

We consider the possible sizes of large sumfree sets contained in the discrete hypercube $\{1,...,n\}^k$, and we determine upper and lower bounds for the maximal size as $n$ becomes large. We also discuss a continuous analogue in which our lower bound remains valid and our upper bound can be strengthened, and we consider the generalization of both problems to $l$-fold-sumfree sets.