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Inverse questions for the large sieve
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Green, Ben and Harper, Adam (2014) Inverse questions for the large sieve. Geometric and Functional Analysis, 24 (4). pp. 1167-1203. doi:10.1007/s00039-014-0288-1 ISSN 1420-8970.
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Official URL: https://doi.org/10.1007/s00039-014-0288-1
Abstract
Suppose that an infinite set A occupies at most 1 / 2 ( p + 1) residue classes modulo p , for every sufficiently large prime p . The squares, or more generally the integer values of any quadratic, are an example of such a set. By the large sieve inequality the number of elements of A that are at most X is O ( X 1 / 2 ), and the quadratic examples show that this is sharp. The simplest form of the inverse large sieve problem asks whether they are the only examples. We prove a variety of results and formulate various conjectures in connection with this problem, including several improvements of the large sieve bound when the residue classes occupied by A have some additive structure. Unfortunately we cannot solve the problem itself.
Item Type: | Journal Article | |||||||||||||||
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Subjects: | Q Science > QA Mathematics | |||||||||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | |||||||||||||||
Library of Congress Subject Headings (LCSH): | Sieves (Mathematics), Inverse problems (Differential equations) | |||||||||||||||
Journal or Publication Title: | Geometric and Functional Analysis | |||||||||||||||
Publisher: | Springer Basel | |||||||||||||||
ISSN: | 1420-8970 | |||||||||||||||
Official Date: | August 2014 | |||||||||||||||
Dates: |
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Volume: | 24 | |||||||||||||||
Number: | 4 | |||||||||||||||
Page Range: | pp. 1167-1203 | |||||||||||||||
DOI: | 10.1007/s00039-014-0288-1 | |||||||||||||||
Status: | Peer Reviewed | |||||||||||||||
Publication Status: | Published | |||||||||||||||
Reuse Statement (publisher, data, author rights): | This is a post-peer-review, pre-copyedit version of an article published in Geometric and Functional Analysis. The final authenticated version is available online at: https://doi.org/10.1007/s00039-014-0288-1 | |||||||||||||||
Access rights to Published version: | Restricted or Subscription Access | |||||||||||||||
Date of first compliant deposit: | 21 August 2018 | |||||||||||||||
Date of first compliant Open Access: | 21 August 2018 | |||||||||||||||
RIOXX Funder/Project Grant: |
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