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[This post, which is based on an ongoing discussion with Alex Russell and Ravi Sundaram, contains some unpublished results.]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

Continue reading “When does the Discrete Fourier Transform Matrix have Nonsingular Submatrices?”

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.

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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.

Continue reading “Impagliazzo’s Hardcore Lemma: a Proof”