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[This post, which is based on an ongoing discussion with Alex Russell and Ravi Sundaram, contains some unpublished results.]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 $\lim_{N\rightarrow \infty}{f(x)/g(x) = 0}$.

When does the Discrete Fourier Transform Matrix have Nonsingular Submatrices?

I am studying a coding theory problem. The question is this:

Open Question: Is there a prime $p$ and a positive integer $d$ such that all submatrices of the $p\times p$ Discrete Fourier Transform matrix over the field $GF(2^d)$ are nonsingular?

Currently, I have only counterexamples: Let $d$ be the degree of the smallest extension over $GF(2)$ which contains a nontrivial $p$th root of unity. Then, I know a lot of primes $p$ for which the matrix $V$ has a singular submatrix.

In this post, I am going to show a failed attempt to answer this question using the results in this paper by Evra, Kowalski, and Lubotzky.

Bounding the Supremum of a Gaussian Process: Talagrand’s Generic Chaining (Part 1)

This post is part of a series which answers a certain question about the supremum of a Gaussian process. I am going to write, as I have understood, a proof given in Chapter 1 of the book “Generic Chaining” by Michel Talagrand. I recommend the reader to take a look at the excellent posts by James Lee on this matter. (I am a beginner, James Lee is a master.)

Let $(T,d)$ be a finite metric space. Let $\{X_t\}$ be a Gaussian process where each $X_t$ is a zero-mean Gaussian random variable. The distance between two points $s,t\in T$ is the square-root of the covariance between $X_s$ and $X_t$. In this post, we are interested in upper-bounding $Q$.

Question: How large can the quantity $Q := \mathbb{E} \sup_{t\in T} X_t$ be?

In this post we are going to prove the following fact:

$\boxed{\displaystyle \mathbb{E}\sup_{t\in T}X_t \leq O(1)\cdot \sup_{t\in T}\sum_{n\geq 1}{2^{n/2}d(t,T_{n-1})} ,}$

where $(t, A)$ is the distance between the point $X_t$ from the set $A$, and $\{T_i\}$ is a specific sequence of sets with $T_i\subset T$. Constructions of these sets will be discussed in a subsequent post.

Impagliazzo’s Hardcore Lemma: a Proof

Informally speaking, Impagliazzo’s hardcore lemma says that if a boolean function is “hard to compute on average” by small circuits, then there exists a set of inputs on which the same function is “extremely hard to compute on average” by slightly smaller circuits.

In this post, I am going to explain how I understand the proof of the hardcore lemma presented in the Arora-Barak complexity book (here). Because the formal proof can be found in the book I intend to write in an informal way. I think some subtleties are involved in turning the context of the lemma into a suitable two-player zero-sum game. Doing so enables one to use von Neumann’s minimax theorem to effectively “exchange the quantifiers” in the contrapositive statement of the lemma. Although the Arora-Barak proof mentions these subtleties, I am going to explore these in more detail and in a more accessible way for a beginner like me.