scholarly journals Continued fractions with low complexity: transcendence measures and quadratic approximation

2012 ◽  
Vol 148 (3) ◽  
pp. 718-750 ◽  
Author(s):  
Yann Bugeaud
Keyword(s):  
Lower Bound ◽  
Continued Fractions ◽  
Low Complexity ◽  
Quadratic Approximation ◽  
Real Numbers ◽  
Partial Quotients ◽  
Block Complexity

AbstractWe establish measures of non-quadraticity and transcendence measures for real numbers whose sequence of partial quotients has sublinear block complexity. The main new ingredient is an improvement of Liouville’s inequality giving a lower bound for the distance between two distinct quadratic real numbers. Furthermore, we discuss the gap between Mahler’s exponent w2 and Koksma’s exponent w*2.

scholarly journals On the frequency of partial quotients of regular continued fractions

2009 ◽  
Vol 148 (1) ◽  
pp. 179-192 ◽  
Author(s):  
AI-HUA FAN ◽  
LINGMIN LIAO ◽  
JI-HUA MA
Keyword(s):  
Variational Principle ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Real Numbers ◽  
Hausdorff Dimensions ◽  
Partial Quotients ◽  
Continued Fraction Expansions

AbstractWe consider sets of real numbers in [0, 1) with prescribed frequencies of partial quotients in their regular continued fraction expansions. It is shown that the Hausdorff dimensions of these sets, always bounded from below by 1/2, are given by a modified variational principle.

A note on approximation efficiency and partial quotients of Engel continued fractions

2017 ◽  
Vol 13 (09) ◽  
pp. 2433-2443
Author(s):  
Hui Hu ◽  
Yueli Yu ◽  
Yanfen Zhao
Keyword(s):  
Hausdorff Dimension ◽  
Growth Rates ◽  
Continued Fractions ◽  
Real Numbers ◽  
Best Approximations ◽  
Hausdorff Dimensions ◽  
Partial Quotients ◽  
Set Of Points

We consider the efficiency of approximating real numbers by their convergents of Engel continued fractions (ECF). Specifically, we estimate the Hausdorff dimension of the set of points whose ECF-convergents are the best approximations infinitely often. We also obtain the Hausdorff dimensions of the Jarnik-like set and the related sets defined by some growth rates of partial quotients in ECF expansions.

scholarly journals Low-complexity feed-forward carrier phase estimation for M-ary QAM based on phase search acceleration by quadratic approximation

2015 ◽  
Vol 23 (15) ◽  
pp. 19142 ◽  
Author(s):  
Meng Xiang ◽  
Songnian Fu ◽  
Lei Deng ◽  
Ming Tang ◽  
Perry Shum ◽  
...  
Keyword(s):  
Carrier Phase ◽  
Low Complexity ◽  
Quadratic Approximation ◽  
Phase Estimation ◽  
Feed Forward

scholarly journals ON INTEGER SEQUENCES GENERATED BY LINEAR MAPS

2009 ◽  
Vol 51 (2) ◽  
pp. 243-252
Author(s):  
ARTŪRAS DUBICKAS
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Real Number ◽  
Constant Independent ◽  
Integer Sequences ◽  
Real Numbers ◽  
Complexity Function ◽  
Linear Maps ◽  
Coprime Integers ◽  
Positive Integers

AbstractLetx0<x1<x2< ⋅⋅⋅ be an increasing sequence of positive integers given by the formulaxn=⌊βxn−1+ γ⌋ forn=1, 2, 3, . . ., where β > 1 and γ are real numbers andx0is a positive integer. We describe the conditions on integersbd, . . .,b0, not all zero, and on a real number β > 1 under which the sequence of integerswn=bdxn+d+ ⋅⋅⋅ +b0xn,n=0, 1, 2, . . ., is bounded by a constant independent ofn. The conditions under which this sequence can be ultimately periodic are also described. Finally, we prove a lower bound on the complexity function of the sequenceqxn+1−pxn∈ {0, 1, . . .,q−1},n=0, 1, 2, . . ., wherex0is a positive integer,p>q> 1 are coprime integers andxn=⌈pxn−1/q⌉ forn=1, 2, 3, . . . A similar speculative result concerning the complexity of the sequence of alternatives (F:x↦x/2 orS:x↦(3x+1)/2) in the 3x+1 problem is also given.

scholarly journals Renewal-type limit theorem for continued fractions with even partial quotients

2009 ◽  
Vol 29 (5) ◽  
pp. 1451-1478 ◽  
Author(s):  
FRANCESCO CELLAROSI
Keyword(s):  
Limit Theorem ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Type Theorem ◽  
Natural Extension ◽  
Gauss Map ◽  
Continued Fraction Expansion ◽  
Special Flow ◽  
Partial Quotients ◽  
Continued Fraction Expansions

AbstractWe prove the existence of the limiting distribution for the sequence of denominators generated by continued fraction expansions with even partial quotients, which were introduced by Schweiger [Continued fractions with odd and even partial quotients. Arbeitsberichte Math. Institut Universtät Salzburg4 (1982), 59–70; On the approximation by continues fractions with odd and even partial quotients. Arbeitsberichte Math. Institut Universtät Salzburg1–2 (1984), 105–114] and studied also by Kraaikamp and Lopes [The theta group and the continued fraction expansion with even partial quotients. Geom. Dedicata59(3) (1996), 293–333]. Our main result is proven following the strategy used by Sinai and Ulcigrai [Renewal-type limit theorem for the Gauss map and continued fractions. Ergod. Th. & Dynam. Sys.28 (2008), 643–655] in their proof of a similar renewal-type theorem for Euclidean continued fraction expansions and the Gauss map. The main steps in our proof are the construction of a natural extension of a Gauss-like map and the proof of mixing of a related special flow.

scholarly journals CONTINUED FRACTIONS WITH BOUNDED PARTIAL QUOTIENTS

2002 ◽  
Vol 45 (3) ◽  
pp. 653-671 ◽  
Author(s):  
J. L. Davison
Keyword(s):  
Rate Of Convergence ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Irrational Number ◽  
Subject Classification ◽  
Specific Sequence ◽  
Secondary 11B37 ◽  
Quadratic Irrational ◽  
Infinite Word ◽  
Partial Quotients

AbstractPrecise bounds are given for the quantity$$ L(\alpha)=\frac{\limsup_{m\rightarrow\infty}(1/m)\ln q_m}{\liminf_{m\rightarrow\infty}(1/m)\ln q_m}, $$where $(q_m)$ is the classical sequence of denominators of convergents to the continued fraction $\alpha=[0,u_1,u_2,\dots]$ and $(u_m)$ is assumed bounded, with a distribution.If the infinite word $\bm{u}=u_1u_2\dots$ has arbitrarily large instances of segment repetition at or near the beginning of the word, then we quantify this property by means of a number $\gamma$, called the segment-repetition factor.If $\alpha$ is not a quadratic irrational, then we produce a specific sequence of quadratic irrational approximations to $\alpha$, the rate of convergence given in terms of $L$ and $\gamma$. As an application, we demonstrate the transcendence of some continued fractions, a typical one being of the form $[0,u_1,u_2,\dots]$ with $u_m=1+\lfloor m\theta\rfloor\Mod n$, $n\geq2$, and $\theta$ an irrational number which satisfies any of a given set of conditions.AMS 2000 Mathematics subject classification: Primary 11A55. Secondary 11B37

Quadratic approximation in ℚp

2014 ◽  
Vol 11 (01) ◽  
pp. 193-209 ◽  
Author(s):  
Yann Bugeaud ◽  
Tomislav Pejković
Keyword(s):  
Prime Number ◽  
Continued Fractions ◽  
Quadratic Approximation ◽  
Almost All

Let p be a prime number. Let w2 and [Formula: see text] denote the exponents of approximation defined by Mahler and Koksma, respectively, in their classifications of p-adic numbers. It is well-known that every p-adic number ξ satisfies [Formula: see text], with [Formula: see text] for almost all ξ. By means of Schneider's continued fractions, we give explicit examples of p-adic numbers ξ for which the function [Formula: see text] takes any prescribed value in the interval (0, 1].

scholarly journals Homomorphic Comparison for Point Numbers with User-Controllable Precision and Its Applications

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 788
Author(s):  
Heewon Chung ◽  
Myungsun Kim ◽  
Ahmad Al Badawi ◽  
Khin Mi Mi Aung ◽  
Bharadwaj Veeravalli
Keyword(s):  
Continued Fraction ◽  
Homomorphic Encryption ◽  
Data Types ◽  
Real Numbers ◽  
The Public ◽  
Public Keys ◽  
Real World Applications ◽  
User Data ◽  
Partial Quotients ◽  
And Storage

This work is mainly interested in ensuring users’ privacy in asymmetric computing, such as cloud computing. In particular, because lots of user data are expressed in non-integer data types, privacy-enhanced applications built on fully homomorphic encryption (FHE) must support real-valued comparisons due to the ubiquity of real numbers in real-world applications. However, as FHE schemes operate in specific domains, such as that of congruent integers, most FHE-based solutions focus only on homomorphic comparisons of integers. Attempts to overcome this barrier can be grouped into two classes. Given point numbers in the form of approximate real numbers, one class of solution uses a special-purpose encoding to represent the point numbers, whereas the other class constructs a dedicated FHE scheme to encrypt point numbers directly. The solutions in the former class may provide depth-efficient arithmetic (i.e., logarithmic depth in the size of the data), but not depth-efficient comparisons between FHE-encrypted point numbers. The second class may avoid this problem, but it requires the precision of point numbers to be determined before the FHE setup is run. Thus, the precision of the data cannot be controlled once the setup is complete. Furthermore, because the precision accuracy is closely related to the sizes of the encryption parameters, increasing the precision of point numbers results in increasing the sizes of the FHE parameters, which increases the sizes of the public keys and ciphertexts, incurring more expensive computation and storage. Unfortunately, this problem also occurs in many of the proposals that fall into the first class. In this work, we are interested in depth-efficient comparison over FHE-encrypted point numbers. In particular, we focus on enabling the precision of point numbers to be manipulated after the system parameters of the underlying FHE scheme are determined, and even after the point numbers are encrypted. To this end, we encode point numbers in continued fraction (CF) form. Therefore, our work lies in the first class of solutions, except that our CF-based approach allows depth-efficient homomorphic comparisons (more precisely, the complexity of the comparison is O ( log κ + log n ) for a number of partial quotients n and their bit length κ , which is normally small) while allowing users to determine the precision of the encrypted point numbers when running their applications. We develop several useful applications (e.g., sorting) that leverage our CF-based homomorphic comparisons.

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