We obtain positive and negative results concerning lacunary discrete maximal operators defined by dilations of sufficiently nonsingular hypersurfaces arising from Diophantine equations in many variables. Our negative results show that this problem differs substantially from that of lacunary discrete maximal operators defined along a nonsingular hypersurface. Our positive results are improvements over bounds for the corresponding full maximal functions which were initially studied by Magyar. In order to obtain positive results, we use an interpolation technique of the second author to reduce problem to a maximal function of main terms. The main terms take the shape of those introduced in work of the first author, which is a more localized version of the main terms that appear in work of Magyar. The main ingredient of this paper is a new bound on the main terms near $\ell^1$. For our negative results we generalize an argument of Zienkiewicz.

中文翻译：

---

由 Birch-Magyar 平均值给出的 Lacunary 极大函数的界限

我们获得了关于由许多变量中的丢番图方程产生的足够非奇异超曲面的膨胀定义的空缺离散最大算子的正面和负面结果。我们的负面结果表明，这个问题与沿非奇异超曲面定义的空缺离散最大算子的问题有很大不同。我们的积极结果是对 Magyar 最初研究的相应完整极大函数的改进。为了获得积极的结果，我们使用第二作者的插值技术将问题简化为主要项的最大函数。主要术语采用第一作者作品中介绍的形式，这是 Magyar 作品中出现的主要术语的更加本地化的版本。本文的主要内容是在 $\ell^1$ 附近的主要项上的新界限。对于我们的否定结果，我们概括了 Zienkiewicz 的论点。