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The discrete spherical averages over a family of sparse sequences

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Abstract

We initiate the study of the ℓp(ℤd)-boundedness of the arithmetic spherical maximal function over sparse sequences. We state a folklore conjecture for lacunary sequences, a key example of Zienkiewicz and prove new bounds for a family of sparse sequences that achieves the endpoint of the Magyar-Stein-Wainger theorem for the full discrete spherical maximal function in [MSW02]. Perhaps our most interesting result is the boundedness of a discrete spherical maximal function in ℤ4 over an infinite, albeit sparse, set of radii. Our methods include the Kloosterman refinement for the Fourier transform of the spherical measure (introduced in [Mag07]) and Weil bounds for Kloosterman sums which are utilized by a new further decomposition of spherical measure.

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Acknowledgements

The author would like to thank Lillian Pierce for discussions on the arithmetic lacunary spherical maximal function and for pointing out a critical mistake in a previous version of this paper, and his advisor, Elias Stein, for introducing him to the problem. The author would also like to thank Roger Heath-Brown for a discussion on the limitations of the circle method, Peter Sarnak for discussions regarding Kloosterman sums and Jim Wright for explaining aspects of the continuous lacunary spherical maximal function. And a special thanks to Lutz Helfrich, James Maynard and Kaisa Matomäki at the Hausdorff Center for Mathematics’s ENFANT and ELEFANT conferences in July 2014 for pointing out the family of sequences used in Theorem 2.

Finally the author thanks Jonathan Hickman and Marina Iliopoulou for their feedback on a previous draft of this paper.

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  1. School of Mathematics, The University of Edinburgh, James Clerk Maxwell Building, The King’s Buildings, Peter Guthrie Tait Road, Edinburgh, EH9 3FD, Scotland, UK

    Kevin Hughes

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Correspondence to Kevin Hughes.

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Hughes, K. The discrete spherical averages over a family of sparse sequences. JAMA 138, 1–21 (2019). https://doi.org/10.1007/s11854-019-0020-z

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  • DOI: https://doi.org/10.1007/s11854-019-0020-z

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