Some observations on Erdős matrices

Abstract

While characterizing 2 times 2 Erdős matrices is trivial, it is only recently that a complete characterization (up to equivalence) of 3 times 3 Erdős matrices was obtained. The result due to Bouthat, Mashreghi and Morneau-Guérin (2024) appears to have revived interest in the question raised by Erdős. Two new results were obtained in the article under evaluation. First, it is shown that, for n greater or equal to 4, there are (with some equivalence) a finite number of n times n Erdős matrices. If the demonstration provides an upper bound for the number of such matrices that is anything but sharp, it has the advantage of providing an algorithmic procedure for testing and generating Erdős matrices. By way of example, the author revisits the case of dimension n = 3. Although the results obtained in doing so are not original, the method used to obtain them has the advantage of being more revealing of the underlying dynamics. Second, a question raised by Bouthat, Mashreghi and Morneau-Guérin is resolved. It is shown that every Erdős matrix has rational entries. Finally, another interesting aspect of this article (which is independent of the other two) is that it proposes a natural generalization of Erdős' question