Gaussian primes in small boxes

Question

The best unconditional result bounding prime gaps is due to Baker, Harman and Pintz, and states that for any sufficiently large $n$, the interval $$[n,n+Cn^{0.525}]$$ contains a prime, for some constant $C>0$. I was curious whether an analogous result holds in the Gaussian integers.

Basic Question: What is the slowest growing monotonic function $F$ such that for any sufficiently large $n,m>0,$ the box $$[n,n+F(n)]\times [m,m+F(m)]$$ contains a Gaussian prime?

We'll assume that $m\asymp n$ since the possibility of $m$ being significantly smaller than $n$ may cause significant additional problems that detract from what I want to ask about here (notably if $m$ is fixed, this becomes a variant of Landau's problem).

For this reason, I am particularly interested in the case $m=n$, as any method that applies here likely also applies when $m\asymp n$. The case $m=n$ takes the following simple form:

Main Question: What is the slowest growing function $F$ such that for sufficiently large $n$, there always exists $$a,b\in [n,n+F(n)]$$ with $a^2+b^2$ prime?

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Remark 1: There are results for Gaussian primes in narrow sectors, which can viewed as an analogue of prime gaps in the Gaussian integers. That is, very small sectors in radial coordinates where the angle tends to $0$ quite quickly as the radius tends to infinity will still contain Gaussian primes. See this Mathoverflow answer, and this paper of Harman and Lewis. I am wondering if such a result holds in Cartesian coordinates.

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Remark 2: In more generality, we can ask about representations by norm forms. Let $f\in\mathbb{Z}[X]$ be some monic irreducible polynomial of degree $n$ with root $\omega\in\mathbb{C}$, let $K=\mathbb{Q}(\omega)$ and let $N(\vec{x})$ for $\vec{x}\in\mathbb{Z}^n$ be defined by $$N(\vec{x})=N(x_1,\dots,x_n)=N_{K/\mathbb{Q}}\left(\sum_{i=1}^{n} x_i \omega^{i-1}\right).$$ We know how to count the number of $\vec{x}\in[1,X]^n$ such that $N(\vec{x})$ is prime, and in fact, it is even known that a right order of magnitude count can be obtained when $\vec{x}$ is restricted to the significantly smaller box $[1,X]^{3n/4}\times [0]^{n/4}$ (see Theorem 1.1 here). The generalized question is then:

General Version: What is the slowest growing function $F=F_K$, which likely depends on the dimension $n$, but possibly more specifically on the number field $K$, such that for sufficiently large $X$ there exists $$\vec{x}\in[X,X+F(X)]^n$$ such that $N(\vec{x})$ is prime?

Answer 1

I think that Bill Duke proves a "near miss" to what you seek in his Ph.D. thesis (http://matwbn.icm.edu.pl/ksiazki/aa/aa52/aa5231.pdf). The idea proceeds as follows (roughly speaking). Let $K$ be a number field of degree $n_K$ over $\mathbb{Q}$. For $\delta\geq0$, let $N(x_1,\ldots,x_{n_K})$ be defined as in your Remark 2, and fix an integer $1\leq m\leq n_K$. Let

$\mathscr{P}_{\delta,N}=\{p$ prime$\colon p=N(x_1,\ldots,x_{n_K}),~|x_i|\leq p^{\frac{1}{n_K}-\delta}$ for $i\neq m\}$,

and define $\pi_{\delta,N}(x)=\#\{p\leq x\colon x\in \mathscr{P}_{\delta,N}\}$. Duke proved that if $0\leq\delta<\frac{1}{3n_K}$, then

$\pi_{\delta,N}(x)\asymp \frac{x^{1-(n_K-1)\delta}}{\log x}$.

By a minor tweak to Duke's proof, we can change $p\leq x$ to $x<p\leq 2x$, in which case we have that each $x_i$ lies in an interval roughly of the shape $[-x^{\frac{1}{n_K}-\delta},x^{\frac{1}{n_K}-\delta}]$ (except when $i=m$, where $m$ was chosen at the onset). The proof is similar in spirit to Hecke's proof of the equidistribution of Gaussian primes in sectors (see, for example, Theorem 5.36 of Iwaniec and Kowalski), but in order to ensure that the $x_i$ lie in "short intervals", Duke required a zero density estimate for Hecke grossencharacters.

I suspect that Duke's ideas could be adapted to yield something closer to what you want.