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A fourth power discrepancy mean

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Abstract

Let \(S\) be a bounded closed convex plane set with sufficiently smooth boundary curve. The area of \(S\) is the number of integer points in \(S\) minus a correction, the local discrepancy. Kendall’s classic paper introduced the Fourier transform of the local discrepancy and found the best possible mean square estimate. We obtain a corresponding fourth power estimate, valid merely under a \(C^2\) smoothness condition.

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Authors and Affiliations

  1. School of Mathematics, Cardiff University, 23 Senghennydd Road, Cardiff, CF24 4YH, Wales, UK

    M. N. Huxley

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  1. M. N. Huxley
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Correspondence to M. N. Huxley.

Additional information

Communicated by J. Schoißengeier.

In memory of D.G. Kendall 1918–2007 and E. Hlawka 1916–2009.

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Huxley, M.N. A fourth power discrepancy mean. Monatsh Math 173, 231–238 (2014). https://doi.org/10.1007/s00605-013-0558-2

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  • DOI: https://doi.org/10.1007/s00605-013-0558-2

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