Large solution-free sets via combinatorial degenerations

Abstract

Consider the linear form $L(x,y,z,w) = 3x+y-2z-2w$. For a positive integer $N$, denote by $r_L(N)$ the largest size of a subset of $\{1,2,\dots,N\}$ that avoids nontrivial solutions to $L = 0$. We show that $r_L(N) = Ω(N^{0.5608687})$, improving the lower bound for Problem 16 in Green's list of open problems. Our proof uses the method of combinatorial degenerations to turn a finite certificate into large solution-free sets. In fact, we can improve Ruzsa's lower bound of $N^{1/2-o(1)}$ for many four-variable equations. Consider $L(x,y,z,w) = ax+by-cz-dw$ with $a,b,c,d\in \mathbb Z_{>0}$, $a+b=c+d$, $\{a,b\} \neq \{c,d\}$ and $abcd$ not a square. We show that there is $\varepsilon_L > 0$ such that $r_L(N) = Ω_L(N^{1/2 + \varepsilon_L})$. Furthermore, we show that for primitive translation-invariant linear forms $L$ in $s$ variables, the lower bound $Ω_s(N^{1/(s-1)})$ coming from a greedy construction is never optimal.