Near optimal thresholds for existence of dilated configurations in ${\mathbb{F}}_q^d$

Pablo Bhowmik, Firdavs Rakhmonov

Let $E\subset {\mathbb{F}}_q^d$ and $\Vert \cdot \Vert :{\mathbb{F}}_q^d\to {\mathbb{F}}_q$ defined as $\Vert \alpha \Vert :={\alpha }_1^2+\cdots +{\alpha }_d^2$ if $\alpha =({\alpha }_1,\dots ,{\alpha }_d)\in {\mathbb{F}}_q^d$, wher…