Affine Copies of Three-Point Patterns in Sets of Integers

Abstract

Let $P=\{0,a,b\}$, where $0<a<b$ and $\gcd (a,b)=1$. For a finite set $A\subset \mathbb{Z}$, let $M_P^{+}(A)$ count the copies $x,x+ad,x+bd\in A$ with $d>0$, and let $M_P(A)$ count the copies with any $d\ne 0$. We prove that every such three-point pattern other than the arithmetic progression $\{0,1,2\}$ satisfies \[ M_P^+(A)\le \frac{99}{400}|A|^2+O(|A|), \qquad M_P(A)\le \frac{13}{28}|A|^2+O_P(|A|). \] For the particular pattern $P=\{0,1,3\}$ -- the subject of a question raised by Ganguly and recorded as Problem 24 in Green's list of open problems -- we sharpen the bound allowing both signs of the dilation to \[ M_{\{0,1,3\}}(A)\le \frac{47}{122}|A|^2+O(|A|). \]