Number of lattice point inside a circle in general position.

Question

There is a lot of literature for number-of-lattice-points-inside-a-circle when the circle is centred at the origin, see for example the Gauss circle problem. But what happens if the requirement for the centre is dropped and instead we consider a circle in general position and want bounds on the minimum and maximum number of lattice points?

In a question on another site, experimental results for $r=\sqrt{23}$ show the maximum is 76, more than the 69 for the circle centred at the origin. The opposite occurs for $r=\sqrt{20}$, and the maximum number of points, 69, occurs for a circle centre the origin, but the minimum number is 58.

Are there any results for this wider problem?

Answer 1

A first, simple estimate can be obtained as follows by simple geometry and subset relation of circles:

Let $C(r,P)$ denote the circle of radius $r$ around point $P$, and let $\#C$ be the number of lattice points in $C$.

The first observation is that we can take $P$ mod 1 and the number of lattice points does not change because the lattice is $\Bbb Z^2$ and $x-(x\bmod 1)\in\Bbb Z$:

$$\#C(r,P) = \#C(r,P \bmod 1) \tag 1$$

where for a real number $x$ we chose remainders of $x\bmod 1$ in $[-1/2, 1/2]$; and for a point $P=(x,y)$ we set $P \bmod 1 := (x\bmod 1, y\bmod1)$. Thus $P\bmod 1\in[-1/2,1/2]^2$. Moreover we have

$$C(r,P) \subseteq C(r+|P|,0) \tag 2$$

and thus

$$\#C(r,P) = \#C(r,P \bmod 1)\leqslant \#C(r + |P \bmod 1|,0)$$

where $|P \bmod 1|\leqslant 1/\sqrt 2$. A bound from below follows from

$$C(r-|P|,0) \subseteq C(r,P) \tag 3$$

where the circle on the left side decays to the empty set for $|P|>r$. Taking it together:

$$\#C(r-|P \mod 1|,0) ~\leqslant~ \#C(r,P) ~\leqslant~ \#C(r+|P\bmod 1|, 0) \tag 4$$

and the less strict version

$$\#C(r-1/\sqrt2,0) ~\leqslant~ \#C(r,P) ~\leqslant~ \#C(r+1/\sqrt2, 0) \tag 5$$

For example, when you have a circle of radius $r$ around $P=(3.3, 6.6)$, then $|P\bmod 1|= \sqrt{0.3^2+0.4^2}=1/2$, and the number of points is bounded between the number of points of the circles or radius $r\pm1/2$ around the origin.