Monochromatic sums equal to products near zero

TL;DR

This paper extends Hindman's monochromatic sum-product result from natural numbers to dense subsemigroups of positive reals near zero, revealing new combinatorial properties in this setting.

Contribution

It generalizes Hindman's theorem to dense subsemigroups of positive reals near zero, broadening the scope of sum-product combinatorics.

Findings

Existence of monochromatic solutions near zero in dense subsemigroups

Extension of sum-product results to new algebraic structures

New combinatorial properties established in dense subsemigroups

Abstract

Hindman proved that, whenever the set $\mathbb{N}$ of naturals is finitely colored, there must exist non-constant monochromatic solution of the equation $a+b=cd$. In this paper we extend this result for dense subsemigroups of $((0, \infty), +)$ to near zero.