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Extensions of Autocorrelation Inequalities with Applications to Additive Combinatorics

Fish, Sara and King, Dylan and Miller, Steven J. (2020) Extensions of Autocorrelation Inequalities with Applications to Additive Combinatorics. Bulletin of the Australian Mathematical Society, 102 (3). pp. 451-461. ISSN 0004-9727. https://resolver.caltech.edu/CaltechAUTHORS:20201216-152007704

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Abstract

Barnard and Steinerberger [‘Three convolution inequalities on the real line with connections to additive combinatorics’, Preprint, 2019, arXiv:1903.08731] established the autocorrelation inequality Min_(0≤t≤1)∫_Rf(x)f(x+t) dx ≤ 0.411||f||²L¹, for fϵL¹(R), where the constant 0.4110.411 cannot be replaced by 0.370.37. In addition to being interesting and important in their own right, inequalities such as these have applications in additive combinatorics. We show that for f to be extremal for this inequality, we must have max min_(x₁∈R 0≤t≤1)[f(x₁−t)+f(x₁+t)] ≤ min_max(x₂∈ R0≤t≤1)[f(x₂−t)+f(x₂+t)]. Our central technique for deriving this result is local perturbation of f to increase the value of the autocorrelation, while leaving ||f||L¹|| unchanged. These perturbation methods can be extended to examine a more general notion of autocorrelation. Let d, n∈Z⁺, f∈L¹, A be a d×n matrix with real entries and columns a_i for 1≤i≤n and C be a constant. For a broad class of matrices A, we prove necessary conditions for f to extremise autocorrelation inequalities of the form Min_(t∈ [0,1]^d)∫R∏_(i=1)^n f(x+t⋅a_i)dx≤C||f||^nL¹.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1017/s000497272000026xDOIArticle
https://arxiv.org/abs/2001.02326arXivDiscussion Paper
ORCID:
AuthorORCID
Miller, Steven J.0000-0003-4904-4605
Additional Information:© 2020 Australian Mathematical Publishing Association Inc. Received 24 January 2020; accepted 15 February 2020; first published online 8 April 2020. This work was supported by NSF grants DMS1659037 and DMS1561945, Wake Forest University and Williams College. We thank Charles Devlin VI and Stefan Steinerberger for helpful conversations about this problem.
Funders:
Funding AgencyGrant Number
NSFDMS-1659037
NSFDMS-1561945
Wake Forest UniversityUNSPECIFIED
Williams CollegeUNSPECIFIED
Subject Keywords:autocorrelation, integral inequality
Issue or Number:3
Classification Code:2010 Mathematics subject classification: primary 26D10; secondary 39B62.
Record Number:CaltechAUTHORS:20201216-152007704
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20201216-152007704
Official Citation:FISH, S., KING, D., & MILLER, S. (2020). Extensions of Autocorrelation Inequalities with Applications to Additive Combinatorics. Bulletin of the Australian Mathematical Society, 102(3), 451-461. doi:10.1017/S000497272000026X
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:107131
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:16 Dec 2020 23:46
Last Modified:16 Dec 2020 23:46

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