scholarly journals Higher Order Fourier Analysis

2012 ◽  
Author(s):  
Terence Tao
Keyword(s):  
Fourier Analysis ◽  
Higher Order

Nil–Bohr sets of integers

2009 ◽  
Vol 31 (1) ◽  
pp. 113-142 ◽  
Author(s):  
BERNARD HOST ◽  
BRYNA KRA
Keyword(s):  
Fourier Analysis ◽  
Ergodic Theory ◽  
Higher Order ◽  
Additive Combinatorics ◽  
Additive Structure ◽  
Upper Density ◽  
Bounded Gaps

AbstractWe study relations between subsets of integers that are large, where large can be interpreted in terms of size (such as a set of positive upper density or a set with bounded gaps) or in terms of additive structure (such as a Bohr set). Bohr sets are fundamentally abelian in nature and are linked to Fourier analysis. Recently it has become apparent that a higher order, non-abelian, Fourier analysis plays a role both in additive combinatorics and in ergodic theory. Here we introduce a higher-order version of Bohr sets and give various properties of these objects, generalizing results of Bergelson, Furstenberg, and Weiss.

scholarly journals Limits of Boolean Functions on $\mathbb{F}_p^n$

10.37236/4445 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Hamed Hatami ◽  
Pooya Hatami ◽  
James Hirst
Keyword(s):  
Fourier Analysis ◽  
Boolean Functions ◽  
Abelian Groups ◽  
Decomposition Theorem ◽  
Higher Order ◽  
Affine Subspace ◽  
Distribution Theorem ◽  
Gowers Norms ◽  
Sequence Of Functions ◽  
Convergent Sequences

We study sequences of functions of the form $\mathbb{F}_p^n \to \{0,1\}$ for varying $n$, and define a notion of convergence based on the induced distributions from restricting the functions to a random affine subspace. Using a decomposition theorem   and a recently proven equi-distribution theorem from higher order Fourier analysis, we prove that the limits of such convergent sequences can be represented by certain measurable functions.  We also show that every such limit object arises as the limit of some sequence of functions. These results are in the spirit of similar results which have been developed for limits of graph sequences. A more general, albeit substantially more sophisticated, limit object was recently constructed by Balázs Szegedy [Gowers norms, regularization and limits of functions on abelian groups. 2010. arXiv:1010.6211].

scholarly journals Higher-order Fourier Analysis and Applications

2019 ◽  
Author(s):  
Hamed Hatami ◽  
Pooya Hatami ◽  
Shachar Lovett
Keyword(s):  
Fourier Analysis ◽  
Higher Order

Higher Order Fourier Analysis for Multiple Species Plasma

PIERS Online ◽  
2006 ◽  
Vol 2 (4) ◽  
pp. 409-411
Author(s):  
Dirk K. Callebaut ◽  
Geoffrey K. Karugila ◽  
Ahmed. H. Khater
Keyword(s):  
Fourier Analysis ◽  
Higher Order ◽  
Multiple Species
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