Three Questions...

Recording 3 questions recently posted at MathOverflow:

1. On Geometry of Numbers:

https://mathoverflow.net/questions/381542/on-circles-and-ellipses-drawn-on-an-infinite-planar-square-lattice

Thanks to Profs. Noam Elkies and Alexei Ustinov for sharing their insights.

Just recording the main questions: Consider a square lattice on the plane formed by points with integer coordinates:

- Given any positive integer n, can we always find a sufficiently large circle drawn on the plane that passes through at least n lattice points? Can such circles be found if the center is not to be a lattice point? What if we require the circle to pass through exactly n lattice points?

- Above qn has a natural restatement if instead of circles, we look at ellipses (either all with a given eccentricity e or with e that can be freely chosen). The ellipses need not be axis parallel.

- And what can one say if the lattice of points has as unit cell not a square but a general parallelogram? And what happens in 3D and higher dimensions?

2. On Zonogons:

https://mathoverflow.net/questions/381781/partitions-of-convex-planar-regions-into-zonogons

A discussion is on...

Here are the main questions: A zonogon is a centrally symmetric convex polygon.

- Are there convex non-zonogons that can be partitioned into a finite number of (convex) zonogons?

- Same as 1 with the pieces allowed to be nonconvex but centrally symmetric polygons.

3. On Congruent Partitions:

https://mathoverflow.net/questions/381091/on-congruent-partitions-of-planar-regions

Given any integer n, any rectangular region or any sector of a disc (including the full disk as a boundary case) can be cut into n mutually congruent pieces - by equally spaced parallel lines and lines radiating from a point at equal angular spacing respectively.

Intuitively, this property generalizes to some deformations of the rectangle which continue to allow partition into n congruent pieces by mutually parallel and equally spaced polylines or curves for any n and for some deformations of a sector which can be congruent partitioned by mutually congruent curves radiating from a single point. .

Question: Are there any other classes of planar regions which can be cut into n mutually congruent regions for any n? The answer seems negative, but is t

here a proof? Note: Analogously, in 3D, one readily has parallelopipeds and suitable slices of regions with axial symmetry (sphere, torus, cone...) which can obviously be cut into n mutually congruent 3D regions for any n.