Given $d, N \in \mathbb{N}$, we define $\mathfrak{C}_d(N)$ to be the number
of pairs of $d\times d$ matrices $A,B$ with entries in $[-N,N] \cap \mathbb{Z}$
such that $AB = BA$. We prove that $$ N^{10} \ll \mathfrak{C}_3(N) \ll
N^{10},$$ thus confirming a speculation of Browning-Sawin-Wang. We further
establish that $$ \mathfrak{C}_2(N) = K(2N+1)^5 (1 + o(1)),$$ where $K>0$ is an
explicit constant. Our methods are completely elementary and rely on upper
bounds of the correct order for restricted divisor correlations with high
uniformity.

Questo articolo esplora i giri e le loro implicazioni.

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