scholarly journals On the Mixed Littlewood Conjecture and continued fractions in quadratic fields

2016 ◽  
Vol 162 ◽  
pp. 1-10
Author(s):  
Paloma Bengoechea ◽  
Evgeniy Zorin
Keyword(s):  
Continued Fractions ◽  
Quadratic Fields ◽  
Littlewood Conjecture

scholarly journals Continued fractions and class number two

2001 ◽  
Vol 27 (9) ◽  
pp. 565-571
Author(s):  
Richard A. Mollin
Keyword(s):  
Continued Fractions ◽  
Class Number ◽  
Quadratic Field ◽  
Ideal Theory ◽  
Quadratic Polynomial ◽  
Quadratic Fields ◽  
Real Quadratic Field ◽  
Real Quadratic Fields ◽  
New Class ◽  
Real Quadratic

We use the theory of continued fractions in conjunction with ideal theory (often called the infrastructure) in real quadratic fields to give new class number 2 criteria and link this to a canonical norm-induced quadratic polynomial. By doing so, this provides a real quadratic field analogue of the well-known result by Hendy (1974) for complex quadratic fields. We illustrate with several examples.

Class Numbers of Real Quadratic Fields, Continued Fractions, Reduced Ideals, Prime-Producing Quadratic Polynomials and Quadratic Residue Covers

1992 ◽  
Vol 44 (4) ◽  
pp. 824-842 ◽  
Author(s):  
S. Louboutin ◽  
R. A. Mollin ◽  
H. C. Williams
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Quadratic Residue ◽  
Quadratic Fields ◽  
Class Numbers ◽  
Continued Fraction Expansion ◽  
Quadratic Polynomials ◽  
Real Quadratic Fields ◽  
The Relationship ◽  
Real Quadratic

AbstractIn this paper we consider the relationship between real quadratic fields, their class numbers and the continued fraction expansion of related ideals, as well as the prime-producing capacity of certain canonical quadratic polynomials. This continues and extends work in [10]–[31] and is related to work in [3]–[4].

scholarly journals On determining certain real quadratic fields with class number one and relating this property to continued fractions and primality properties

1991 ◽  
Vol 124 ◽  
pp. 157-180 ◽  
Author(s):  
Eugène Dubois ◽  
Claude Levesque
Keyword(s):  
Continued Fractions ◽  
Class Number ◽  
Quadratic Fields ◽  
Imaginary Quadratic Fields ◽  
Real Quadratic Fields ◽  
Real Quadratic

Thanks to K. Heegner [He], A. Baker [Ba] and H. Stark [S], we know that there are nine imaginary quadratic fields of class number one. Gauss conjectured that there are infinitely many real quadratic fields of class number one, but the conjecture is still open.

scholarly journals Higher-rank Bohr sets and multiplicative diophantine approximation

2019 ◽  
Vol 155 (11) ◽  
pp. 2214-2233 ◽  
Author(s):  
Sam Chow ◽  
Niclas Technau
Keyword(s):  
Diophantine Approximation ◽  
Continued Fractions ◽  
Structural Theory ◽  
Real Numbers ◽  
Successive Minima ◽  
Arbitrary Rank ◽  
Littlewood Conjecture ◽  
Higher Rank ◽  
Almost All

Gallagher’s theorem is a sharpening and extension of the Littlewood conjecture that holds for almost all tuples of real numbers. We provide a fibre refinement, solving a problem posed by Beresnevich, Haynes and Velani in 2015. Hitherto, this was only known on the plane, as previous approaches relied heavily on the theory of continued fractions. Using reduced successive minima in lieu of continued fractions, we develop the structural theory of Bohr sets of arbitrary rank, in the context of diophantine approximation. In addition, we generalise the theory and result to the inhomogeneous setting. To deal with this inhomogeneity, we employ diophantine transference inequalities in lieu of the three distance theorem.

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