Bounds on largest possible square in sum of two squares

Question

Suppose we are given integers $k,c$ such that $k=1+c^2$.

Let $n$ be an odd integer and suppose that $k^n=a_i^2+b_i^2$ for distinct positive integers $a_i<b_i$ and $i\le d$. That is, there are $d$ different ways to express $k^n$ as a sum of two squares.

For instance, $(a_1,b_1)=(k^{(n-1)/2},ck^{(n-1)/2})$ is a valid pair.

What can be said about $\max b_i$? Are there good bounds (as a function of $c,n$) on the magnitude of largest possible square when writing an integer as a sum of two squares?

---

Addendum: As mentioned below, we can rephrase this problem in terms of the irrationality measure of $\arctan(1/c)/\pi$. I'm having a lot of trouble finding results on irrationality measures of values of inverse trigonometric functions in general, but I could be missing something.

Answer 1

Rather than discuss $\max b_{i}$, I'll discuss the equivalent question of bounding $\min a_{i}$. The ABC conjecture implies that for all $\epsilon > 0$, $\min a_{i} \gg (c^{2}+1)^{n/2 - 1 - \epsilon}$. This is because if we have a solution to $a^{2} + b^{2} = (c^{2}+1)^{n}$ with $a \ll (c^{2} + 1)^{n/2 - 1 - \epsilon}$, set $A = a^{2}$, $B = b^{2}$ and $C = (c^{2}+1)^{n}$. Assume for simplicity that $\gcd(a,b) = 1$. (It doesn't change much if $\gcd(a,b) > 1$.) Then $C \ll {\rm rad}(ABC)^{1+\delta}$ for all $\delta > 0$. This gives $$ (c^{2}+1)^{n} < (ab(c^{2}+1))^{1+\delta} \ll ((c^{2}+1)^{n/2 - 1 - \epsilon} (c^{2}+1)^{n/2} (c^{2}+1))^{1+\delta} = ((c^{2}+1)^{n-\epsilon})^{1+\delta} $$ which is a contradiction if $\delta < \frac{\epsilon}{n-\epsilon}$.

For $n = 3$, it is possible to construct a sequence of integers $c$ which getse close to this bound. In particular, let $$ c_{k} = \frac{(2 + \sqrt{3})^{k} - (2 - \sqrt{3})^{k}}{\sqrt{3}} \quad a_{k} = \frac{(2 + \sqrt{3})^{k} + (2 - \sqrt{3})^{k}}{2}. $$ It is easy to see that $a_{k}, c_{k} \in \mathbb{Z}$, $c_{k}$ is even, and a somewhat tedious calculation shows that $$ (c_{k}^{2} + 1)^{3} = a_{k}^{2} + \left(c_{k}^{3} + \frac{3}{2} c_{k}\right)^{2}. $$ In particular $\min a_{i} \leq a_{k} \approx \frac{\sqrt{3}}{2} c_{k}$.

One could hope to generalize this construction by finding $c_{k}^{2} + 1 = d_{k}^{2} + e_{k}^{2}$, and $\frac{e_{k}}{c_{k}} \approx \sin\left(\frac{\pi}{2n}\right)$. Letting $\frac{e_{k}}{\sqrt{d_{k}^{2} + e_{k}^{2}}} = \sin(\theta_{k})$, this makes $$ (c_{k} + i)^{n} = (d_{k} + ie_{k})^{n} \approx (c_{k}^{2} + 1)^{n/2} \left(\cos(\theta_{k}) + i \sin(\theta_{k})\right)^{n} = (c^{2} + 1)^{n/2} \left(\cos(n \theta_{k}) + i \sin(n \theta_{k}\right)) $$ and $\sin(n \theta_{k}) \approx \sin(\pi/2) = 1$. This boils down to finding points on the hyperboloid $x^{2}+y^{2} = z^{2} + 1$ that lie close to the line $y = \sin\left(\frac{\pi}{2n}\right) z$.