scholarly journals Gowers norms for the Thue–Morse and Rudin–Shapiro sequences

2019 ◽  
Vol 69 (4) ◽  
pp. 1897-1913 ◽  
Author(s):  
Jakub Konieczny
Keyword(s):  
Gowers Norms

scholarly journals An equivalence between inverse sumset theorems and inverse conjectures for theU3norm

2010 ◽  
Vol 149 (1) ◽  
pp. 1-19 ◽  
Author(s):  
BEN GREEN ◽  
TERENCE TAO
Keyword(s):  
Finite Field ◽  
Abelian Groups ◽  
Field Vector ◽  
Vector Spaces ◽  
Cyclic Groups ◽  
Inverse Results ◽  
Inverse Theorems ◽  
Gowers Norms

AbstractWe establish a correspondence betweeninverse sumset theorems(which can be viewed as classifications of approximate (abelian) groups) andinverse theorems for the Gowers norms(which can be viewed as classifications of approximate polynomials). In particular, we show that the inverse sumset theorems of Freĭman type are equivalent to the known inverse results for the GowersU3norms, and moreover that the conjectured polynomial strengthening of the former is also equivalent to the polynomial strengthening of the latter. We establish this equivalence in two model settings, namely that of the finite field vector spaces2n, and of the cyclic groups ℤ/Nℤ.In both cases the argument involves clarifying the structure of certain types ofapproximate homomorphism.

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 Energies and Structure of Additive Sets

10.37236/4369 ◽  
2014 ◽  
Vol 21 (3) ◽  
Author(s):  
Shkredov Ilya
Keyword(s):  
Disjoint Union ◽  
Difference Set ◽  
Full Description ◽  
Random Structure ◽  
Large Subset ◽  
Gowers Norms

In this paper we prove that any sumset or difference set has large $\textsf{E}_3$ energy. Also, we give a full description of families of sets having critical relations between some kind of energies such as $\textsf{E}_k$, $\textsf{T}_k$ and Gowers norms. In particular, we give criteria for a set to be a set of the form $H\dotplus \Lambda$, where $H+H$ is small and $\Lambda$ has "random structure",set equal to a disjoint union of sets $H_j$ each with small doubling,set having a large subset $A'$ with $2A'$ equal to a set with small doubling and $|A'+A'| \approx |A|^4 / \textsf{E}(A)$.

scholarly journals Parallelepipeds, nilpotent groups and Gowers norms

2008 ◽  
Vol 136 (3) ◽  
pp. 405-437 ◽  
Author(s):  
Bernard Host ◽  
Bryna Kra
Keyword(s):  
Nilpotent Groups ◽  
Gowers Norms

On the gowers norms of certain functions

2012 ◽  
Vol 92 (3-4) ◽  
pp. 554-569
Author(s):  
I. D. Shkredov
Keyword(s):  
Gowers Norms

scholarly journals An inverse theorem for Gowers norms of trace functions over Fp

2013 ◽  
Vol 155 (2) ◽  
pp. 277-295 ◽  
Author(s):  
ÉTIENNE FOUVRY ◽  
EMMANUEL KOWALSKI ◽  
PHILIPPE MICHEL
Keyword(s):  
The Other ◽  
Inverse Theorem ◽  
Random Functions ◽  
Other Hand ◽  
The One ◽  
Gowers Norms

AbstractWe study the Gowers uniformity norms of functions over Z/pZ which are trace functions of ℓ-adic sheaves. On the one hand, we establish a strong inverse theorem for these functions, and on the other hand this gives many explicit examples of functions with Gowers norms of size comparable to that of “random” functions.

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