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An asymptotic version of the prime power conjecture for perfect difference sets

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Abstract

We show that the number of positive integers \(n\le N\) such that \({\mathbb {Z}}/(n^2+n+1){\mathbb {Z}}\) contains a perfect difference set is asymptotically \(\frac{N}{\log {N}}\).

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Acknowledgements

The author thanks Ben Green and Kannan Soundararajan for helpful conversations and comments on earlier drafts of this paper and the anonymous referee for many useful suggestions. The author is supported by the NSF Mathematical Sciences Postdoctoral Research Fellowship Program under Grant No. DMS-1903038

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  1. Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK

    Sarah Peluse

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  1. Sarah Peluse
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Correspondence to Sarah Peluse.

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Communicated by Kannan Soundararajan.

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Peluse, S. An asymptotic version of the prime power conjecture for perfect difference sets. Math. Ann. 380, 1387–1425 (2021). https://doi.org/10.1007/s00208-021-02188-5

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