The following question is well known:
Consider representations of a given integer as sums of two squares, i.e. solutions to $a^2 + b^2 = n$ in $a,b\in\mathbb Z$ with $n$ fixed. As $n \to \infty$, are the normalized points $\left(\frac a {\sqrt n}, \frac b {\sqrt n}\right)$ uniformly distributed on the unit circle?
An equivalent formulation is given by:
Denote by $\Delta(n)$ the discrepancy $$\Delta(n) = \sup_{\Gamma\text{ arc on }\mathcal S^1} \left|\frac {\text{number of points for }n\text{ in }\Gamma}{\text{number of points for }n} - \frac {\text{Length}(\Gamma)} {2\pi}\right|$$ Then as $n \to \infty$, is it true that $\Delta(n) \to 0$?
The answer is "it depends on what you mean by $n \to \infty$". Obviously, for many $n$ there are no points at all or a small number of points. Even if we require the number of points to grow to infinity, a counterexample was constructed by J. Cilleruelo in The distribution of the lattice points on circles, J. Number Theory 43 (1993), no. 2, p. 198-202.
An answer in the positive direction is given by Erdős and Hall in On the angular distribution of Gaussian integers with fixed norm, Discrete Math 200 (1999), p. 87-94. The formulation is something like this (which I reproduce here in a simplified fashion):
For "almost all" integers $n \le x$ that are representable as a sum of two squares, we have $\Delta(n) \le \log ^{-\kappa} x$, where $\kappa > 0$ is an absolute constant.
A similar result (with essentially the same analysis) was given earlier by Kátai and Környei in On the distribution of lattice points on circles, Ann. Univ. Sci. Budapest., Sect. Math. 19 (1977), p. 87-91.
I have two questions:
The analysis done by Erdős and Hall and by Kátai and Környei is based on an averaging argument. This means they say nothing about which $n$ give low discrepancy, only that most $n$ do. Is there a known result that gives criteria on $n$ to ensure low discrepancy? (In contrast, Cilleruelo's counterexample construction gives a criterion to ensure high discrepancy.)
It seems odd to me that this result is so recent (1999 or even 1977): It has been done in higher dimensions, which seems a lot harder, in 1959 (Pommerenke), and in this MathOverflow question a reference to a similar problem was given from 1920.
Is this "folklore question" - if so, what is "new" about the more recent results? Could you give me a reference for the simplest way to solve the problem if what I want is to show that for "many sequences of integers" $n \to \infty$ (preferably, with description) we have $\Delta(n) \to 0$, and I don't care about the speed of convergence?
On the other hand, if this indeed is a new result - how come it wasn't known in the 1950s?