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Research Spending & Results

Award Detail

Awardee:UNIVERSITY OF MAINE SYSTEM
Doing Business As Name:University of Maine
PD/PI:
  • Brandon Hanson
  • (416) 899-3209
  • brandon.w.hanson@gmail.com
Award Date:05/25/2021
Estimated Total Award Amount: $ 95,398
Funds Obligated to Date: $ 89,924
  • FY 2020=$89,924
Start Date:06/01/2021
End Date:06/30/2023
Transaction Type:Grant
Agency:NSF
Awarding Agency Code:4900
Funding Agency Code:4900
CFDA Number:47.049
Primary Program Source:040100 NSF RESEARCH & RELATED ACTIVIT
Award Title or Description:Arithmetic structure and distribution
Federal Award ID Number:2135200
DUNS ID:186875787
Parent DUNS ID:071750426
Program:Combinatorics
Program Officer:
  • Stefaan De Winter
  • (703) 292-2599
  • sgdewint@nsf.gov

Awardee Location

Street:5717 Corbett Hall
City:ORONO
State:ME
ZIP:04469-5717
County:Orono
Country:US
Awardee Cong. District:02

Primary Place of Performance

Organization Name:University of Maine
Street:5717 Corbett Hall
City:ORONO
State:ME
ZIP:04469-5717
County:Orono
Country:US
Cong. District:02

Abstract at Time of Award

Many questions in arithmetic can be phrased in terms of the superposition of waves by way of the Fourier Transform. A basic question is "What do the frequencies of these waves tell us about their interference and vice versa?". The interference of the waves has to do with the magnitude of their superposition, and the concentration of the interference can be made large by choosing the frequencies according to a strict pattern. One of the questions pursued by the PI is the converse: are these patterns necessary to achieve a very concentrated superposition. This is called the Inverse Littlewood Problem. The second question under investigation is an uncertainty principle. Suppose we have two families of waves that can be used to decompose a signal, and these two families are very incompatible. This incompatibility means we expect that a simple wave from one family looks much more complicated as a superposition from the second. A specific instance of this is whether a multiplicative character (a very simple wave in one family) can be decomposed as an additive convolution (a fairly simple superposition in another family). This is called Sarkozy's Problem. There is a long history of comparing the distribution and arithmetic structure of sequences in number theory. Classically, the connection is observed by way of the Fourier transform. Towards a better understanding of this phenomenon, the PI plans to build upon recent results that relate the dimension of a finite subset of a lattice to estimates on the L^1-norm of its associated Fourier series. This area of investigation began with a conjecture of Littlewood on how small the L^1 -norm of the Fourier transform of a finite set of integers could be. The problem was resolved independently by Konyagin and McGehee-Pigno-Smith in the 1980s. Recently, questions have turned to classifying the extremizers in this problem, and the PI has recently proved that they must be, in an appropriate sense, low-dimensional. In this project he will develop the method further, by weakening hypotheses and applying these results to problems in arithmetic reliant the L^1 norm. In the finite field setting, the PI and Petridis have made substantial progress on a problem of Sarkozy concerning sumsets and the quadratic residues. They have resolved the conjecture for almost all primes. The PI will develop the ideas further so as to resolve the conjecture in earnest. Beyond this, he will adapt the method to other problems, which in turn will lead to progress on Vinogradov’s conjecture on the least non-residue. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

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