# Large finite subsets of Euclidean space with no isosceles (or approximately isosceles) triangles

Here’s a question in combinatorial geometry which feels very much like other questions I’m familiar with but which I can’t see how to get a hold of. I’ll actually propose two different questions on the same theme. The continuous question is the one I’d really like to know the answer to, but the discrete question feels more “mainstream” and tractable and the two are clearly related.

Discrete Question:

Let $$SS$$ be a subset of the $$N×NN \times N$$ grid such that no three distinct points of $$SS$$ (including collinear points) form an isosceles triangle. How big can $$|S||S|$$ be? Can it be as large as $$N2−ϵN^{2-\epsilon}$$?

Continuous Question:

For $$β>0\beta > 0$$, we say a triangle in Euclidean space is $$β\beta$$-isosceles if some vertex is within distance $$β\beta$$ of the line equidistant between the other two vertices.

Let $$SS$$ be a finite subset of $$[0,1]d[0,1]^d$$ such that no three distinct points of $$SS$$ form a $$β\beta$$-isosceles triangle. What is the minimum possible value of the Hausdorff distance between $$SS$$ and $$[0,1]d[0,1]^d$$? In other words, what is the largest $$α\alpha$$ such that there must be a ball of radius $$α\alpha$$ containing no points of $$SS$$?

This is a question I was going to pose as a challenge in a paper I’m writing arising from a problem in data science. I realized it might be a good idea to check with people here as to whether anything is already known about it!

Of course, the 1-dimensional versions of these questions have to do with sets with no 3-term arithmetic progressions, which are the subject of a large literature. But I don’t immediately see how to leverage those results to say anything about these problems, except to say that the answer to the discrete problem has to be $$o(N2)o(N^2)$$. Is there a version of Behrend’s construction that applies to the higher-dimensional case?