Block complexity and idempotent Schur multipliers

We call a matrix blocky if, up to row and column permutations, it can be obtained from an identity matrix by repeatedly applying one of the following operations: duplicating a row, duplicating a column, or adding a zero row or column. Blocky matrices are precisely the boolean matrices that are contractive when considered as Schur multipliers. It is conjectured that any boolean matrix with Schur multiplier norm at most $γ$ is expressible as a signed sum \begin{equation*}A = \sum_{i=1}^L \pm B_i\end{equation*} for some blocky matrices $B_i$, where $L$ depends only on $γ$. This conjecture is an analogue of Green and Sanders’s quantitative version of Cohen’s idempotent theorem. In this paper, we prove bounds on $L$ that are polylogarithmic in the dimension of $A$. Concretely, if $A$ is an $n\times n$ matrix, we show that one may take $L = 2^{O(γ^7)} \log(n)^2$.

💡 Research Summary

The paper investigates a matrix analogue of Cohen’s idempotent theorem and its quantitative version due to Green and Sanders. In the classical setting, a Boolean function on a finite abelian group with small Fourier‑algebra norm can be expressed as a signed sum of a bounded number of coset indicators. Translating this to the language of Schur multipliers, the authors define a Boolean matrix to be blocky if, after permuting rows and columns, it can be obtained from the identity matrix by repeatedly duplicating rows, duplicating columns, or inserting all‑zero rows or columns. Blocky matrices are exactly those Boolean matrices whose Schur‑multiplier norm does not exceed 1.

The central conjecture (Conjecture 1.2) asserts that any Boolean matrix (A) with Schur‑multiplier norm at most (\gamma) can be written as a signed sum of blocky matrices, \