Statistics/Size of Minimal Solution to ax-by=1, Uniform distribution of Skewness, Kloosterman Sums, Poincare Series

The equation $\displaystyle ax-by=1$ has infinitely many solution for any coprime integers $a$ and $b$ . And every solution is given by the translate of a fixed solution by $(b, a)$ . One can find the solution by the gcd Euclidean algorithm applied to $a$ and $b$ . (Thus a solution can be obtained by looking at the last convergent in the continued fraction expansion of $\frac{a}{b}$ .

Question: How does a “minimal” solution look? If $a, b$ are of some size $N$ . what can we say about the relative sizes $\displaystyle \frac{x}{b}$ or $\displaystyle \frac{x^2+y^2}{a^2+b^2}$ ? How are they are distributed?

We are essentially asking for how skewed is the fundamental parallelogram of the integer lattice with a given basis vector $(a, b)$ .

The answer is for random $(a, b)$ of some size, the second basis vector looks completely random– that is the skewness is uniformly distributed.

An analytic way to prove this is using the fact that $x$ is the inverse $\bar a \mod b$ , and hence the uniform distribution reduces to cancellation in Kloosterman sums.

See: Akio Fujii. On a problem of Dinaburg and Sina ̆ı. Proc. Japan Acad. Ser. A Math. Sci., 68(7):198–203, 1992

Because the solution are related to the Euclidean algorithm and continued fractions, one can study the problem using the distributions/dynamics of continued fractions and Gauss map. This gives uniform distribution at a smaller scales than the Kloosterman sums can establish.

See Dimitry Dolgopyat. On the distribution of the minimal solution of a linear Diophantine equation with random coefficients.Funktsional. Anal. i Prilozhen., 28(3):22–34, 95, 1994

Lattice and Modular surface- We can also see that the produce reduces to the uniform distribution of $Re(\gamma z)$ modulo one as $\gamma$ in the modular group varies. To capture this analytically using to Weyl’s criterion, we need to study sums over $\gamma$ closely related to the Poincare series, and establish their analytic properties.

See https://arxiv.org/pdf/0707.3323.pdf

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