Let $N(r)$ be the number of lattice points distance at most $r$ from the origin in $\mathbb{R}$. The Gauss circle problem is a famous problem which is looking to understand the error term $E(r):= N(r)-\pi r^2$. Gauss proved $|E(r)| \leq 2 \sqrt 2 \pi r$.
Let $B_r$ be the ball of radius $r$ around the origin. Consider $$A= \begin{bmatrix} 2 & 1 \\ 1 & 1 \end{bmatrix}$$ which is a matrix in $SL (2, \mathbb Z)$, so it preserves volume and the lattice. Now $A$ has eigenvector with slope $\phi$(the golden ratio), and $A^kB_r$ is an ellipse. For $k$ large(depending on $r$) it is very close to the line with slope $\phi$ with the same number of lattice points. There is nothing particularly special about $A$ except that it has two real eigenvalues and preserves the lattice, but for concreteness I figured I would mention it.
Question: Can we see interesting bounds for $E(r$) by estimating lattice points close to the line $y=\phi x$? (Possibly using the fact $\phi$ is hard to approximate with rationals/continued fractions...) I would like to see if this "dynamical" diophantine approximation could be used to recover Gauss's original bound.
One way to think about the Gauss problem is look at geodesics of length at most $r$ originating from a point on a torus and keeping track of how many times they come back to the starting point.
The rephrasing of the question into how many points are close to a line with slope $\phi$ is basically asking how many times the line "comes close" to the starting point. The main difficulties I imagine are: you have to look at a very long line compared to the radius and, perhaps an even bigger problem, the width of the ellipse around the line will be variable. I feel like there could be some dynamical methods and/or number theoretic estimates which could "get around this" and get a somewhat reasonable bound―maybe worse than Guass's.
I would also like to mention is that if continued fractions could be useful in these estimates then it is likely one could look at train tracks and train track splitting sequences on the torus to do a similar thing. You can look at this pdf What is... A Train Track? by Lee Mosher for an overview on the connection. Perhaps estimating number of curves of length at most $n$ (or something) carried by such train tracks would give interesting estimates.