Timeline for Maximizing the number of lattice points in a circle of radius $r$ placed on a lattice

6 events

when toggle format	what		by	license	comment
Oct 21, 2012 at 21:47	vote	accept	user27203
Oct 21, 2012 at 19:11	comment	added	Michael Biro		Depending on what you mean by small, I think so. Given $r$ and a square, look at all lattice points that are in the circle for some center in the square but not in the circle for a different center. There are $O(r)$ of these points and if you take the arrangement of radius $r$ circles centered at these lattice points, we get $O(r^2)$ faces, where two circles of radius $r$ with centers in the same face in the square contain the same lattice points. So, there are at most $O(r^2)$ different values the function can take, even without looking for peaks, and most small movements won't matter at all.
Oct 21, 2012 at 18:05	comment	added	user27203		Ahem, peaks in terms of the number of lattice points internal to the circle.
Oct 21, 2012 at 18:04	comment	added	user27203		@Michael Biro If a circle of radius $r$ were to execute something like a Brownian motion on a lattice, would we really only expect peaks when its center drifts over a small set of points in the fundamental unit of the Z^2 or A^2 lattices?
Oct 21, 2012 at 17:37	comment	added	user27203		@Michael Biro "I want to say that by symmetry, the only relevant centers are lattice points, midpoints of square edges, and centers of squares..." right, I think that makes a lot of sense... but is there any way to make this statement rigorous?
Oct 21, 2012 at 16:58	history	answered	Michael Biro	CC BY-SA 3.0