A theorem of Macbeath asserts that μ(A + B) ≥ min(1, μ(A) + μ(B)) for any subsets A and B of a finite-dimensional torus. We conjecture that, when the obvious exceptions are excluded, a stronger inequality μ(A + B) ≥ min(1, μ(A) + μ(B) + min(84L?A), μ(B))) holds, and we prove this conjecture under some technical restrictions.
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