Rectangle-to-isosceles-triangle ratios in finite lattice sets

Can the ratio of ordered rectangles to ordered isosceles triangles in a finite planar lattice set be arbitrarily large, counting the allowed degenerate configurations consistently? A quantitative target in [−N,N]² is a ratio at least (log N)^(c/log log log N).

Why it matters and what counts as progress

Why it matters: It tests whether two natural configuration counts admit a universal comparison, and asks for a quantitative construction if they do not.

What counts as progress: Construct lattice sets with unbounded rectangle-to-isosceles-triangle ratio, with consistent ordered and degenerate-count conventions.

How the problem developed

29 Jun 2026 · Reported claim [1]
Jonathan Bennett, Vjekoslav Kovač, Shohei Nakamura, and Itamar Oliveira Introduced explicit finite lattice constructions with many rectangles relative to isosceles triangles, using finite-field constructions and lattice lifting; the paper applies them to the paraboloid Mizohata–Takeuchi estimate.

AI contribution

Rectangles, triangles and Schrödinger waves

Jonathan Bennett, Vjekoslav Kovač, Shohei Nakamura, Itamar Oliveira

Finite planar lattice sets can have an arbitrarily large ratio of ordered rectangles to ordered isosceles triangles, including degenerate configurations. Bennett, Kovač, Nakamura and Oliveira quantify the construction in a box [-N,N]² by a ratio at least (log N)^(c/log log log N), with an analogous cardinality bound.

AI involvementAI-centered

ImportanceNotable

Technical verificationFormal (human)

Human understandingFull