We establish lower bounds on the rank of matrices in which all but the diagonal entries lie in a multiplicative group of small rank. Applying these bounds we show that the distance sets of finite pointsets in $\mathbb {R}^d$ generate high-rank multiplicative groups and that multiplicative groups of small rank cannot contain large sumsets.

中文翻译：

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具有乘法组条目的矩阵的秩

我们在矩阵的秩上建立了下界，其中除了对角线元素之外的所有元素都位于一个小秩的乘法组中。应用这些界限，我们表明 $\mathbb {R}^d$ 中的有限点集的距离集生成高秩乘法群，并且小秩的乘法群不能包含大和集。