What is the status of the Gauss Circle Problem?

Question

For $r > 0$, let $L(r) = \# \{ (x,y) \in \mathbb{Z}^2 \ | \ x^2 + y^2 \leq r^2\}$ be the number of lattice points lying on or inside the standard circle of radius $r$. It is easy to see that $L(r) \sim \pi r^2$ as $r \rightarrow \infty$. The Gauss circle problem is to give the best possible error bounds: put

$E(r) = |L(r) - \pi r^2|$.

Gauss himself gave the elementary bound $E(r) = O(r)$. In 1916 Hardy and Landau showed that it is not the case that $E(r) = O(r^{\frac{1}{2}})$. It is now believed that this is "almost" true: i.e.:

Gauss Circle Conjecture: For every $\epsilon > 0$, $E(r) = O_{\epsilon}(r^{\frac{1}{2}+\epsilon})$.

So far as I know the best published result is a 1993 theorem of Huxley, who shows one may take $\epsilon > \frac{19}{146}$.

(For a little more information, see here, Wayback Machine.)

In early 2007 I was teaching an elementary number theory class when I noticed that Cappell and Shaneson had uploaded a preprint to the arxiv claiming to prove the Gauss Circle Conjecture:

https://arxiv.org/abs/math/0702613

Two more versions were uploaded, the last in July of 2007.

It is now a little more than three years later, and so far as I know the paper has neither been published nor retracted. This seems like a strange state of affairs for an important classical problem. Can someone say what the status of the Gauss Circle Problem is today? Is the argument of Cappell and Shaneson correct? Or is there a known flaw?

Answer 1

When Cappell visited UWM a few months ago, one of my colleagues asked him about the status of the paper. The answer was that it is "still in works", which in plain English, probably, means "having severe problems with some remote hope to fix them". The point is that it contains no idea that hadn't been well-known to experts before it appeared, just an enormous amount of "brute forcing" (which, by the way, makes it very hard to read). Sometimes you can succeed by being just more persistent than others but it doesn't seem to be the case here. The consensus is that the existing methods have already been brought to their extreme and to proceed some fresh idea is required.

Answer 2

This recent arxiv posting by Shaneson claims that one may take $\epsilon > 1153/9750$, improving on Huxley's bound. It also includes the passage

In 2007 Cappell and the author posted a paper on the arXiv claiming to obtain the estimate [in my notation -- PLC] $O(r^{1/2+\epsilon})$. Unfortunately we have not been able to produce an error free version. The present paper shares with that paper the Proposition in section 5 and there is also something there akin to what immediately follows the Proposition.

I guess that's that.

Answer 3

Back in 2007 or so, at a tea I heard a noted expert in the field pooh-poohing it (for instance, sign errors in the Stokes analogue), and he seemed not to want to read any more re-hashes (he had seen more than one from these authors, who seemed to keep changing the argument). This expert is one of those they thank. It was unclear whether he thought their whole idea (to the extent the Intro explained this) was even capable of working. I do not know its submission status.