A sharp dichotomy: the commuting probability for 2x2 real matrices is bounded by 8 times the largest mass in any 2-dimensional subspace, and it is optimal up to a constant.
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On commuting pairs in arbitrary sets of 2x2 matrices
A sharp dichotomy: the commuting probability for 2x2 real matrices is bounded by 8 times the largest mass in any 2-dimensional subspace, and it is optimal up to a constant.