Determinants of matrices related to the Pascal triangle

In this note we prove an that the Pascal matrix modulo $2$ has the property that each of the square sub-matrices laying on the upper border or on the left border has determinant, computed in $\mathbb{Z}$, equal to $1$ or $-1.$ This extends some of the results in [1, 2] on determinants related to the Pascal triangle.

The Pascal triangle matrix has been used in the theory of uniform distribution modulo one to construct sequences of real numbers in the unit interval with smallest possible discrepancy: the first $N$ terms have discrepancy at most $(\log N)/N$ times a constant (see [4] and the references therein). When we restrict to sequences of the form $\{b^{n}x\pmod{1}\}_{n=1,2,3,\ldots}$ for any integer $b$ greater than or equal to $2$ and for real numbers $x$, the smallest exact discrepancy that can be achieved by some $x$ is not known. The question dates back to Korobov in 1956 (cfr. [6]).

Using the Pascal triangle matrix modulo $2$, in [8] M. Levin constructs numbers $x$ such that the sequence $\{b^{n}x\pmod{1}\}_{n=0,1,2,\ldots}$ has discrepancy of the first $N$ terms bounded from above by $(\log N)^{2}/N$ times a constant. Becher and Carton in [3] defined variants of the Pascal triangle matrix modulo $2$ that have the same property of the invertibility of the square sub-matrices laying on the upper or left border. They obtain a family of numbers with the same property as Levin’s. Larcher and Hofer recently showed that for Levin’s number constructed for $b=2$ the discrepancy estimate $(\log N)^{2}/N$ is the best possible (cfr. [5]).

The property that all square matrices in the upper and left border of the Pascal matrix modulo 2 have determinants, computed in $\mathbb{Z}$ equal to $1$ or $-1$ ensures that if these determinants are computed in $\mathbb{Z}/b\mathbb{Z}$, for any $b$, they are also equal to $1$ or $-1$. Thus, indeed, Levin’s method yields numbers $x$ such that the $\{b^{n}x\pmod{1}\}_{n=1,2,\ldots}$ has the small discrepancy property.

The article is organized as follows. In section 2 we introduce some notation, define the infinite matrix $U$ and state the main result (Theorem 1). Section 3 is devoted to its proof. In section 4 we define a whole family of matrices sharing the property of having all its sub-matrices laying on the upper or left border invertible, compute its number and give some examples.

We want to study determinants of sub-matrices of certain infinite matrix. It will be convenient to index the rows and columns with non-negative numbers.

Let $U$ be the infinite matrix whose entry in the $i,j$ position is the remainder when the binomial coefficient $\binom{j}{i}$ is divided by $2.$ Namely

$\displaystyle U_{i,j}={\begin{cases}1&{\text{ if }}\binom{j}{i}\text{ is odd, }\\ 0&{\text{ if }}\binom{j}{i}\text{ is even.}\end{cases}}$

Writing $U_{I,J}$ for the sub-matrix of $U$ corresponding to the rows and columns indexed respectively by the sets $I,J\subseteq\mathbb{Z}_{\geq 0}$ introduce the following notation. The subset of integers $k$ greater than or equal to $m$ but smaller than $n$ is denoted by $[m:n)={\left\{k\in\mathbb{Z}:,m\leq k<n\right\}}.$ The principal minors $U_{[0:n),[0:n)}$ are denoted $U(n).$ For the top-most minors $U_{[0:n),[m:m+n)}$ we write $U_{0,m}(n).$ Finally, $U_{i,j}(n,m)$ stands for $U_{[i:i+n)[j:j+m)}.$

The main result of this article is the following:

Our matrix $U$ appeared in [2] to prove that all symmetric Pascal matrices $\pmod{2}$ have determinant $\pm 1.$

Consider $P$ the infinite matrix²²2it is noted $\overline{P}(\infty)_{2}$ in [2].

$\displaystyle P_{i,j}={\begin{cases}1&{\text{ if }}\binom{j+i}{i}\text{ is odd, }\\ 0&{\text{ if }}\binom{j+i}{i}\text{ is even.}\end{cases}}$

For its $LU$-decomposition we define $L$ as the transpose of $U$ and $D$ as the infinite diagonal with the Thue–Morse sequence ³³3https://oeis.org/A106400. Namely

$\displaystyle D_{i,j}={\begin{cases}1&{\text{ if }}i=j\text{ has an even number of $1$'s in base $2$, }\\ -1&{\text{ if }}i=j\text{ has an odd number of $1$'s in base $2$, }\\ 0&{\text{ if }}i\neq j.\end{cases}}$

Adopting the same notation for sub-matrices as with $U$ we have $P,L$ and $D$ are the infinite tensor product of

$\displaystyle P(2)=\begin{pmatrix}1&1\\ 1&0\end{pmatrix},\qquad L(2)=\begin{pmatrix}1&0\\ 1&1\end{pmatrix}\qquad\text{ and }\qquad D(2)=\begin{pmatrix}1&0\\ 0&-1\end{pmatrix}$

respectively.

Since

$$\begin{pmatrix}1&1\\ 1&0\end{pmatrix}=\begin{pmatrix}1&0\\ 1&1\end{pmatrix}\begin{pmatrix}1&0\\ 0&-1\end{pmatrix}\begin{pmatrix}1&1\\ 0&1\end{pmatrix}$$

we get that $P(n)=L(n)D(n)U(n)$ for all $n\geq 1.$

As $L$ is lower triangular and $D$ is diagonal with only $\pm 1$’s, Theorem 1 implies

The same result applies for infinite matrices having a similar $LU$-decomposition.

By the symmetry of $P$ we deduce that every square sub-matrix laying on the upper or left border of $P$ has determinant $\pm 1.$

We say that a matrix is Pascal-like if every such sub-matrix is invertible. When working over the integers this means having determinant $\pm 1$ so our matrix $P$ is Pascal-like by Corollary1.

Another example over the integers is provided by the honest Pascal matrix $M_{i,j}=\binom{i+j}{j}$ as can be seen by a routine application of Vandermonde determinant and elementary row operations.

More is true according to the following

The author would like to thank Prof. Becher for bringing this problem to his attention in the first place, together with a lot of helpful comments as well.