Approximation of Lebesgue integrals by Riemann sums and lattice points in domains with fractal boundary
The general lattice point problem for ''large'' domains is concerned with a bounded and measurable set \(\mathcal D\subset{\mathbb R}^k\) (with characteristic function \(\chi_\mathcal D\)) which may be submitted to an arbitrary rotation \(\sigma\in SO(k)\), a shift by a vector \(\mathbf t\in{\mathbb R^k/\mathbb Z^k}\), and a dilation (''blowing up'') by a large parameter \(x\in\mathbb R\). In a natural way the ''lattice rest'' is defined as \[ P(x,\sigma,\mathbf t) = \sum_{{\mathbf n}\in{\mathbb Z^k}} \chi_{x \sigma^{-1}\mathcal D - \mathbf t} (\mathbf n) \;-\;x^k \text{ vol}(\mathcal D) . \] (number of lattice points minus volume). For the case that the boundary \(\partial \mathcal D\) of \(\mathcal D\) is smooth, a wealth of classic and more recent estimates for \(P(x,\sigma,\mathbf t)\) are available in the literature. The present paper deals with the case that \(\partial \mathcal D\) possesses \textit{fractal} structure. If \(k-\alpha\) denotes the Minkowski dimension of \(\partial \mathcal D\) (\(0\leq\alpha\leq 1\)), the author's main result reads \[ \left({1\over x} \int_0^x \int_{SO(k)} \int_{\mathbb R^k/\mathbb Z^k} P^2(\xi,\sigma,\mathbf t) d\mathbf t\;d\sigma \;d\xi \right)^{1/2} \ll x^{(k-\alpha)/2} . \] This is a very deep and interesting generalization of classic results for the ''smooth'' case (\(\alpha=1\)) due to \textit{G. H. Hardy} [Proc. Lond. Math. Soc. 15, 192-213 (1916; JFM 46.0260.01)], \textit{D. G. Kendall} [Q. J. Oxf., Ser. 19, 1-26 (1948; Zbl 0031.11201)], \textit{B. Randol} [Trans. Am. Math. Soc. 139, 271-285 (1969; Zbl 0183.26905)], and \textit{A. N. Varchenko} [Funkts. Anal. Prilozh. 17, 1-6 (1983; Zbl 0522.10031)]. The above estimate is obtained as a special case of a general theorem which sheds some light on the question how precise Lebesgue integrals can be approximated by Riemann sums: For a function \(\varphi \in L^1(\mathbb R^k) \cap L^p(\mathbb R^k)\) (\(1\leq p\leq 2\)) let \[ E \varphi(x,\sigma,\mathbf t) = x^{-k} \sum_{\mathbf n\in\mathbb Z^k} \varphi\left({1\over x} \sigma(\mathbf t+\mathbf n)\right) - \int_{\mathbb R^k} \varphi(\mathbf u) d\mathbf u. \] The author establishes a sharp bound for \[ \left({1\over x} \int_x^{2x} \int_{SO(k)} \int_{\mathbb R^k/\mathbb Z^k} \left|E \varphi(x,\sigma,\mathbf t)\right|^p d\mathbf t d\sigma d\xi \right)^{1/2} \] by a somewhat more complicated function of \(x\).
scientific article; zbMATH DE number 1019724
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