In this paper we prove an analogue in the discrete setting of \Bbb Z^d, ofthe spherical maximal theorem for \Bbb R^d. The methods used are two-fold: theapplication of certain "sampling" techniques, and ideas arising in the study ofthe number of representations of an integer as a sum of d squares inparticular, the "circle method". The results we obtained are by necessitylimited to d \ge 5, and moreover the range of p for the L^p estimates differsfrom its analogue in \Bbb R^d.
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机译：在本文中，我们证明了\ Bbb Z ^ d的离散设置中\ Bbb R ^ d的球面最大定理的类似物。所使用的方法有两个方面：某些“采样”技术的应用，以及在研究整数表示形式（尤其是d平方和）时所产生的思想，即“圆法”。我们获得的结果必然限于d \ ge 5，而且L ^ p估计值的p范围不同于\ Bbb R ^ d中的类似值。
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