Let A(n,θ) be the maximum number of times angle θ can be formed by n points. What are the values of A(4,θ)? How about A(n,θ) for larger n? Can you prove your answers?
John Hoffman gave many bounds for A(n,θ). For example, he found that A(2n+1,θ) ≥ n2 and A(2n,θ) ≥ n(n-1) by putting one point at the intersection of two lines and evenly dividing the other points on the two lines.
Hoffman can show that for large n, A(n,π/4) ≥ (1-ε)n2 . I can show the same thing for θ=π/2, using a square grid of points. Hoffman and I conjecture that if 0<θ<π, then log A(n,θ)/ log n → 2.
Hoffman and Devincentis showed that A(n,0)=2n(n-1)(n-2)/6 and A(n,π)=n(n-1)(n-2)/6. No more than two angles in any triangle can be θ=0 and no more than one angle can be θ=π. Arranging points in a line achieves both these bounds.
Generalizing this, one can show A(n,θ) ≤ A(n-1,θ) n / (n-3). To see this, consider all n collections of n-1 points out of n points. In each collection, at most A(n-1,θ) angles θ occur, and each such angle appears in (n-3) collections.
There are some general constructions which provide lower bounds. A(n,2π/n) ≥ n(n-2) by putting the points at the vertices of an regular n-gon. Similarly, if θ ≤ 2π/(n-1), then A(n,θ) ≥ (n-1)(n-2) because we can put the points at equally spaced points on the circumference of a circle.
Here are the best known bounds for n=3, 4 and 5.
Here are the conjectured values of A(n,θ) for some special angles. Can you verify or continue any of these sequences?
|θ \ n||3||4||5||6||7|