Vandermonde Submatrices and Arithmetic Progressions

[This post, which is based on an ongoing discussion with Alex Russell and Ravi Sundaram, contains some unpublished results.]

Currently, we are asking whether all submatrices of the order-p Vandermonde matrix over a finite extension of GF(2) are invertible where p is prime. The answer is “no” in general: there are examples of fields where the Vandermonde matrix has a singular submatrix.

We can ask an easier(?) question, though. What happens if we randomly sample a set of columns and look into submatrices formed by a subset of the sampled columns. With a touch of beautiful insight, Professor Russell has connected Szemeredi’s theorem on arithmetic progressions with this question.

Let AP_k denote an arithmetic progression of length $latek k$. Let [N] := \{1, 2, \cdots, N\} for N \in \mathbb{N}.

The Szemerédi theorem says, any “sufficiently dense” subset S \subset [N] contains infinitely many AP_k for all k \in \mathbb{N}. A finitary version says: Fix your favourite k \in \mathbb{N}, \delta \in [0, 1]. Then,  there exists a natural N := N_{k, \delta} such that if you look any subset S \subset [N] of size at least \delta N, you will find an AP_k. Yet another version says:

Szemerédi’s Theorem. The size of the largest subset S \subset [N] without an AP_k cannot be too large; in particular, it is o(N).

Recall that a function f(x) is o(g) if it grows too slow compared to g(x), so that \lim_{N\rightarrow \infty}{f(x)/g(x) = 0}.

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