On √Q-Distances

A subsetXof a Euclidean space is called a √Q-setifd(x, y)2is a rational for every pairx, yinX.To confirm anN-point √Q-set inRn,how many pairs(x, y)do we need to check, provided that theNpoints are in general position? Lets(n, N)denote the minimum number of pairs we need to check. We show that: , N)=( N/2)forN≤n+2; n, n+3)=(n+3/2)-1; n, n+4)=(n+4/2)-4; -n(n+1)/2≤s(n, N)≤nN-n(n+1)/2+⌈2(N-n-1)/3⌉.