### The Heilbronn Triangle Problem

I recently gave a talk (here is an outline) about the work of K. F. Roth on the famous triangle problem of Heilbronn, and was shocked and saddened to discover that Roth passed away that same day. Terry Tao wrote a great post about some of Roth’s most important results in analytic number theory.

Today I will discuss the general approach of Roth, later refined by Komlos, Pintz, and Szemeredi, in proving upper bounds for the Heilbronn Triangle Problem. At the end of the day, the most interesting step comes from a quasi-orthogonality property between indicator functions of bounded strips in the plane. Let $\tau$ be a pair of points in the plane and let $\phi_{\tau}(w,X)$ be the indicator function of the strip of points at most $w$ away from the line through $\tau$. We cut off these functions outside some convex region, say the unit circle – call this the cutoff region. Then, it is natural to consider the “expansion function”

$\Phi_{\tau}(w,w',X) = \frac{1}{w}\phi_{\tau}(w,X) - \frac{1}{w'}\phi_{\tau}(w',X)$,

which measures the weighted difference between two strips of different widths $w$, $w'$ around the same line. The point is that if we integrate this expansion function against the indicator function $f$ of a given set of points in the plane, then $(\Phi_{\tau}, f)$ counts the change in the density of points when we expand from a strip of width $w$ to a strip of width $w'$.