scholarly journals Mattila–Sjölin Type Functions: A Finite Field Model

2021 ◽  
Author(s):  
Daewoong Cheong ◽  
Doowon Koh ◽  
Thang Pham ◽  
Chun-Yen Shen
Keyword(s):  
Finite Field ◽  
Field Model

scholarly journals On Arithmetic Progressions in Symmetric Sets in Finite Field Model

10.37236/9242 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Jan Hązła
Keyword(s):  
Finite Field ◽  
Product Space ◽  
Field Model ◽  
Arithmetic Progressions ◽  
Qualitative Behavior ◽  
Space Model ◽  
Symmetric Set ◽  
The Difference

We consider two problems regarding arithmetic progressions in symmetric sets in the finite field (product space) model. First, we show that a symmetric set $S \subseteq \mathbb{Z}_q^n$ containing $|S| = \mu \cdot q^n$ elements must contain at least $\delta(q, \mu) \cdot q^n \cdot 2^n$ arithmetic progressions $x, x+d, \ldots, x+(q-1)\cdot d$ such that the difference $d$ is restricted to lie in $\{0,1\}^n$. Second, we show that for prime $p$ a symmetric set $S\subseteq\mathbb{F}_p^n$ with $|S|=\mu\cdot p^n$ elements contains at least $\mu^{C(p)}\cdot p^{2n}$ arithmetic progressions of length $p$. This establishes that the qualitative behavior of longer arithmetic progressions in symmetric sets is the same as for progressions of length three.

scholarly journals Stable arithmetic regularity in the finite field model

2018 ◽  
Vol 51 (1) ◽  
pp. 70-88 ◽  
Author(s):  
C. Terry ◽  
J. Wolf
Keyword(s):  
Finite Field ◽  
Field Model

scholarly journals The Depth Efficacy of Unbounded: Characteristic Finite Field Arithmetic

1988 ◽  
Vol 17 (240) ◽  
Author(s):  
Gudmund Skovbjerg Frandsen ◽  
Carl Sturtivant
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Field Model ◽  
Irreducible Polynomial ◽  
Canonical Function ◽  
Full Generality ◽  
Input Size ◽  
Parallel Arithmetic ◽  
Prime Fields ◽  
Bounded Characteristic

We introduce an arithmetic model of parallel computation. The basic operations are ½ and Š gates over finite fields. Functions computed are unary and increasing input size is modelled by shifting the arithmetic base to a larger field. When only finite fields of bounded characteristic are used, then the above model is fully general for parallel computations in that size and depth of optimal arithmetic solutions are polynomially related to size and depth of general (boolean) solutions. In the case of finite fields of unbounded characteristic, we prove that the existence of a fast parallel (boolean) solution to the problem of powering an integer modulo a prime (and powering a polynomial modulo an irreducible polynomial) in combination with the existence of a fast parallel (arithmetic) solution for the problem of computing a single canonical function, f<em>(x)</em>, in the prime fields, guarantees the full generality of the finite field model of computation. We prove that the function f<em>(x)</em>, has a fast parallel arithmetic solution for any ''shallow'' class of primes, i.e. primes <em>p</em> such that any prime power divisor <em>q</em> of <em>p</em> -1 is bounded in value by a polynomial in log <em>p</em>.

Beta dynamical systems over the formal fields

2014 ◽  
Vol 51 (4) ◽  
pp. 454-465
Author(s):  
Lu-Ming Shen ◽  
Huiping Jing
Keyword(s):  
Dynamical Systems ◽  
Finite Field ◽  
Haar Measure ◽  
Laurent Series ◽  
Formal Laurent Series ◽  
Almost All

Let \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{F}_q ((X^{ - 1} ))$$ \end{document} denote the formal field of all formal Laurent series x = Σ n=ν∞anX−n in an indeterminate X, with coefficients an lying in a given finite field \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{F}_q$$ \end{document}. For any \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\beta \in \mathbb{F}_q ((X^{ - 1} ))$$ \end{document} with deg β > 1, it is known that for almost all \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$x \in \mathbb{F}_q ((X^{ - 1} ))$$ \end{document} (with respect to the Haar measure), x is β-normal. In this paper, we show the inverse direction, i.e., for any x, for almost all \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\beta \in \mathbb{F}_q ((X^{ - 1} ))$$ \end{document}, x is β-normal.

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