Sets with no solutions to $x+y=3z$

TL;DR

This paper establishes an upper bound on the measure of subsets of [0,1] that contain no solutions to the equation x+y=3z, contributing to the understanding of solution-free sets in additive combinatorics.

Contribution

It provides a new upper bound on the measure of sets avoiding solutions to x+y=3z within the interval [0,1].

Findings

Derived an explicit upper bound for measure of solution-free sets.

Advances understanding of additive structures in subsets of real intervals.

Contributes to the theory of solution-free sets in additive combinatorics.

Abstract

This short note gives an upper bound on the measure of sets $A\subset [0,1]$ such that $x+y=3z$ has no solutions in $A$.