In this paper we consider estimating the number of solutions to multiplicative equations in finite fields when the variables run through certain sets with high additive structure. In particular, we consider estimating the multiplicative energy of generalized arithmetic progressions in prime fields and of boxes in arbitrary finite fields. We obtain sharp bounds in more general scenarios than previously known. Our arguments extend some ideas of Konyagin and Bourgain and Chang into new settings.

中文翻译：

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有限域中的一些乘法方程

在本文中，我们考虑在变量通过某些具有高加性结构的集合时估计有限域中乘法方程的解数。特别是，我们考虑估计素域中的广义等差数列和任意有限域中的框的乘法能量。我们在比以前已知的更一般的场景中获得了清晰的界限。我们的论点将 Konyagin、Bourgain 和 Chang 的一些想法扩展到新的环境中。