The quotient set of the quadratic distance set over finite fields
Alex Iosevich, Doowon Koh, Firdavs Rakhmonov ยท Forum Mathematicum ยท 2024
Abstract Let ๐ฝ q d {\mathbb{F}_{q}^{d}} be the d-dimensional vector space over the finite field ๐ฝ q {\mathbb{F}_{q}} with q elements. For each non-zero r in ๐ฝ q {\mathbb{F}_{q}} and E โ ๐ฝ q d {E\subset\mathbb{F}_{q}^{d}} , we define W โข ( r ) {W(r)} as the number of quadruples ( x , y , z , w ) โ E 4 {(x,y,z,w)\in E^{4}} such that Q โข ( x - y ) Q โข ( z - w ) = r {\frac{Q(x-y)}{Q(z-w)}=r} , where Q is a non-degenerate quadratic form in d variables over ๐ฝ q {\mathbb{F}_{q}} . When Q โข ( ฮฑ ) = โ i = 1 d ฮฑ i 2