Maier Phenomena for Gauss Circle Problem

Question

For an arithmetic function $\alpha(n)$, let $S_{\alpha}(x) = \sum_{n \le x} \alpha(n)$. When $\alpha$ is the indicator function of primes, Maier has shown that $$\limsup \frac{S_{\alpha}(x+\Phi(x))-S_{\alpha}(x)}{\Phi(x)/\ln x} > 1 > \liminf \frac{S_{\alpha}(x+\Phi(x))-S_{\alpha}(x)}{\Phi(x)/\ln x}$$ when $\Phi(x)=\ln^a x, a>1$.

Consider $\alpha(n) = r(n) = \# \{ (a,b) \in \mathbb{Z}^2 : n=a^2+b^2 \}$. Gauss showed that $S_{r}(x) =\pi x+O(\sqrt{x})$, and Hardy proved that $$\limsup \frac{S_{r}(x) - \pi x}{x^{1/4}}=\infty, \, \liminf \frac{S_{r}(x) - \pi x}{x^{1/4}}=-\infty.$$

Question 1: What is the `slowest growing' $\Phi(x)$ for which it is expected that $\frac{S_{r}(x+\Phi(x))-S_{r}(x)}{\pi \Phi(x)} \sim 1$?

The main conjecture on the Gauss circle problem implies that we may take $\Phi(x)=x^{1/4+\varepsilon}$, but I am not sure what about `slower' $\Phi$-s.

Question 2: What is known about the "Maier phenomena" for $r(n)$? Namely, what is the `fastest growing' $\Phi(x)<x$ for which

$$\limsup \frac{S_{r}(x+\Phi(x))-S_{r}(x)}{\pi \Phi(x)} > 1 > \liminf \frac{S_{\alpha}(x+\Phi(x))-S_{\alpha}(x)}{\pi \Phi(x)}?$$

Another question is related to Hardy's result.

Question 3: What is know about the set $X_a$ of $x$- for which $\frac{S_r(x)-\pi x}{x^{1/4}} > a$, and the set $Y_a$ of $x$- for which $\frac{S_r(x)-\pi x}{x^{1/4}} < a$?

If we have good information about the gaps in such sets, this might help one to establish a Maier phenomena.

Answer 1

It's not a good idea to ask for the Maier phenomenon for $r(n)$. The reason is that individual values of $r(n)$ can get quite large: for example, by choosing $n$ to be the product of the first several primes that are $1 \pmod 4$, one can make $r(n)$ as large as $\exp(C \log n/\log \log n)$. Therefore the Maier phenomenon as such is trivial here, and one should not expect asymptotics in intervals shorter than $\exp(C \log x/\log \log x)$. I would guess that the asymptotic for $r(n)$ would hold in intervals of size $x^{\epsilon}$, for any fixed $\epsilon >0$.

To get a real analogue of the Maier phenomenon, it is better to consider the indicator function of the set of sums of two squares. This problem was considered by Balog and Wooley (paper in Canadian Math. J.) who established the Maier phenomenon in this context. Then Granville and Soundararajan (paper in Annals) generalized the example of primes and sums of two squares to show that essentially all sifted sequences exhibit a Maier type phenomenon. Roughly speaking, they showed that sifted sequences will either be poorly distributed in short intervals (of Maier type length), or will be poorly distributed in arithmetic progressions to fairly small moduli. Finally, you may also want to look at work of Thorne (paper in J. Number Theory) who extended the work of Granville and Soundararajan to function fields.