A note on Zhona's inverse theorem

1995 ◽  
pp. 081-084
Author(s):  
Durmuş BOZKURT
Keyword(s):  
Inverse Theorem

scholarly journals INVERSE THEOREM IN LP-APPROXIMATION BY MODIFIED SZÄSZ-MIRAKYAN OPERATORS

1995 ◽  
Vol 28 (2) ◽  
pp. 285-292
Author(s):  
Vijay Gupta ◽  
G. S. Srivastava ◽  
T. A. K Sinha
Keyword(s):  
Inverse Theorem ◽  
Lp Approximation

An inverse theorem for additive bases

2016 ◽  
Vol 12 (06) ◽  
pp. 1509-1518 ◽  
Author(s):  
Yongke Qu ◽  
Dongchun Han
Keyword(s):  
Abelian Group ◽  
Proper Subgroup ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Regular Sequence ◽  
Inverse Theorem ◽  
Additive Basis

Let [Formula: see text] be a finite abelian group of order [Formula: see text], and [Formula: see text] be the smallest prime dividing [Formula: see text]. Let [Formula: see text] be a sequence over [Formula: see text]. We say that [Formula: see text] is regular if for every proper subgroup [Formula: see text], [Formula: see text] contains at most [Formula: see text] terms from [Formula: see text]. Let [Formula: see text] be the smallest integer [Formula: see text] such that every regular sequence [Formula: see text] over [Formula: see text] of length [Formula: see text] forms an additive basis of [Formula: see text], i.e. [Formula: see text]. Recently, [Formula: see text] was determined for many abelian groups. In this paper, we determined [Formula: see text] for more abelian groups and characterize the structure of the regular sequence [Formula: see text] over [Formula: see text] of length [Formula: see text] and [Formula: see text].

scholarly journals An inverse theorem for the Gowers U^(s+1)[N]-norm

2012 ◽  
Vol 176 (2) ◽  
pp. 1231-1372 ◽  
Author(s):  
Ben Green ◽  
Terence Tao ◽  
Tamar Ziegler
Keyword(s):  
Inverse Theorem

scholarly journals AN INVERSE THEOREM FOR THE GOWERS $U^3(G)$ NORM

2008 ◽  
Vol 51 (1) ◽  
pp. 73-153 ◽  
Author(s):  
Ben Green ◽  
Terence Tao
Keyword(s):  
Absolute Constant ◽  
Bounded Function ◽  
Additive Group ◽  
Arithmetic Progressions ◽  
Inverse Theorem ◽  
Inner Product ◽  
Arbitrary Group ◽  
Quadratic Phase ◽  
Inverse Theorems ◽  
The Fourier Transform

AbstractThere has been much recent progress in the study of arithmetic progressions in various sets, such as dense subsets of the integers or of the primes. One key tool in these developments has been the sequence of Gowers uniformity norms $U^d(G)$, $d=1,2,3,\dots$, on a finite additive group $G$; in particular, to detect arithmetic progressions of length $k$ in $G$ it is important to know under what circumstances the $U^{k-1}(G)$ norm can be large.The $U^1(G)$ norm is trivial, and the $U^2(G)$ norm can be easily described in terms of the Fourier transform. In this paper we systematically study the $U^3(G)$ norm, defined for any function $f:G\to\mathbb{C}$ on a finite additive group $G$ by the formula\begin{multline*} \qquad\|f\|_{U^3(G)}:=|G|^{-4}\sum_{x,a,b,c\in G}(f(x)\overline{f(x+a)f(x+b)f(x+c)}f(x+a+b) \\ \times f(x+b+c)f(x+c+a)\overline{f(x+a+b+c)})^{1/8}.\qquad \end{multline*}We give an inverse theorem for the $U^3(G)$ norm on an arbitrary group $G$. In the finite-field case $G=\mathbb{F}_5^n$ we show that a bounded function $f:G\to\mathbb{C}$ has large $U^3(G)$ norm if and only if it has a large inner product with a function $e(\phi)$, where $e(x):=\mathrm{e}^{2\pi\ri x}$ and $\phi:\mathbb{F}_5^n\to\mathbb{R}/\mathbb{Z}$ is a quadratic phase function. In a general $G$ the statement is more complicated: the phase $\phi$ is quadratic only locally on a Bohr neighbourhood in $G$.As an application we extend Gowers's proof of Szemerédi's theorem for progressions of length four to arbitrary abelian $G$. More precisely, writing $r_4(G)$ for the size of the largest $A\subseteq G$ which does not contain a progression of length four, we prove that$$ r_4(G)\ll|G|(\log\log|G|)^{-c}, $$where $c$ is an absolute constant.We also discuss links between our ideas and recent results of Host, Kra and Ziegler in ergodic theory.In future papers we will apply variants of our inverse theorems to obtain an asymptotic for the number of quadruples $p_1\ltp_2\ltp_3\ltp_4\leq N$ of primes in arithmetic progression, and to obtain significantly stronger bounds for $r_4(G)$.

On approximation in Weighted Orlicz spaces

2012 ◽  
Vol 62 (1) ◽  
Author(s):  
Ali Guven ◽  
Daniyal Israfilov
Keyword(s):  
Approximation Theory ◽  
Orlicz Spaces ◽  
Trigonometric Approximation ◽  
Inverse Theorem ◽  
Lipschitz Classes

AbstractAn inverse theorem of the trigonometric approximation theory in Weighted Orlicz spaces is proved and the constructive characterization of the generalized Lipschitz classes defined in these spaces is obtained.

scholarly journals An Inverse Theorem for the Uniformity Seminorms Associated with the Action of $${{\mathbb {F}^{\infty}_{p}}}$$

2010 ◽  
Vol 19 (6) ◽  
pp. 1539-1596 ◽  
Author(s):  
Vitaly Bergelson ◽  
Terence Tao ◽  
Tamar Ziegler
Keyword(s):  
Inverse Theorem
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