14
$\begingroup$

Most (if not all) references I read about the Gauss Circle Problem that proves a bound below $O(R^{2/3})$ reduces the GCP to the Dirichlet Divisor Problem by the well known expression of $r_2(n)$, the number of ways of writing a natural number $n$ as the sum of two squares. My question is then, what happens if the circle is not centered at the origin (or any lattice point)? In this case it seems that there is no number theoretical exact formula to reduce one problem to the other, and we have to tackle the GCP directly. What are the recent results in this direction?

Addendum: I vaguely recall reading somewhere about the same problem with the circle replaced by a uniformly convex planar domain, in which case the exponent is not as good, but still better than $O(R^{2/3})$. But now I'm unable to find any reference to it, though, so I added a "reference request" tag.

$\endgroup$
  • 3
    $\begingroup$ I don't think the GCP is reduced the the DDP. Instead, the GCP is equivalent to a problem quite similar to the DDP. The reason is simple and you mention it as well: $r_2(n)$ is a divisor sum. $\endgroup$ – GH from MO Dec 29 '15 at 20:56
  • $\begingroup$ @GHfromMO Maybe I didn't read in detail, but if the center of the circle is not a lattice point it is not that easy to relate it to divisor sums, right? $\endgroup$ – Fan Zheng Dec 29 '15 at 22:08
  • $\begingroup$ Consider rotating the offset circle around the lattice point closest to its center (or look at lattice centered concentric inscribed and circumscribed circles). My feeling is that the offset problem differs from the original problem by a divisor sum and an amount that is easily bounded. Gerhard "Or I'm Talking In Circles" Paseman, 2015.12.29 $\endgroup$ – Gerhard Paseman Dec 29 '15 at 22:19
  • $\begingroup$ Related MO question: "Lattice points on the boundary of an ellipse", whose answer cites Bombieri-Pila. $\endgroup$ – Joseph O'Rourke Dec 30 '15 at 2:39
15
$\begingroup$

You can write a similar expression to the usual formula for the remainder term in the circle problem by using the Poisson summation formula. The shift of the center of the circle simply means that one gets a variant of the usual formula for the remainder term modified by suitable exponential terms.

Indeed Huxley has considered a more general problem and obtained analogs of what was known for the usual divisor problem. Huxley considers a closed convex curve $C$ enclosing an area $A$, and the dilate $MC$ of $C$ by a factor $M$. Place this dilate in any way you like (translation or rotation) on the coordinate plane, and count the lattice points inside it. Then under suitable regularity assumptions on the boundary curve $C$, Huxley obtains estimates for the difference between the number of lattice points and the expected number $AM^2$ where $A$ is the area enclosed by $C$. His bounds depend on the original shape $C$, but not on the embedding of $MC$ in the plane. The results are in three papers by him Exponential Sums and Lattice points, I, 2, and 3 (developing a method of Bombieri, Iwaniec and Mozzochi) and also mentioned in a recent survey, see Huxley, which has further references. The strongest result for the translated circle of radius $R$ has an error term of $R^{0.6298\ldots}$ (see page 593 of the third paper).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.