Uniform congruence schemes

1980 ◽  
Vol 10 (1) ◽  
pp. 176-188 ◽  
Author(s):  
E. Fried ◽  
G. Grätzer ◽  
R. Quackenbush
Keyword(s):  
Uniform Congruence

scholarly journals Uniform congruence counting for Schottky semigroups in SL2(𝐙)

2019 ◽  
Vol 2019 (753) ◽  
pp. 89-135 ◽  
Author(s):  
Michael Magee ◽  
Hee Oh ◽  
Dale Winter
Keyword(s):  
Hausdorff Dimension ◽  
Positive Integer ◽  
Continued Fractions ◽  
Euclidean Norm ◽  
Limit Set ◽  
The Family ◽  
Prime Factors ◽  
Uniform Congruence

AbstractLet Γ be a Schottky semigroup in {\mathrm{SL}_{2}(\mathbf{Z})}, and for {q\in\mathbf{N}}, let{\Gamma(q):=\{\gamma\in\Gamma:\gamma=e~{}(\mathrm{mod}~{}q)\}}be its congruence subsemigroup of level q. Let δ denote the Hausdorff dimension of the limit set of Γ. We prove the following uniform congruence counting theorem with respect to the family of Euclidean norm balls {B_{R}} in {M_{2}(\mathbf{R})} of radius R: for all positive integer q with no small prime factors,\#(\Gamma(q)\cap B_{R})=c_{\Gamma}\frac{R^{2\delta}}{\#(\mathrm{SL}_{2}(% \mathbf{Z}/q\mathbf{Z}))}+O(q^{C}R^{2\delta-\epsilon})as {R\to\infty} for some {c_{\Gamma}>0,C>0,\epsilon>0} which are independent of q. Our technique also applies to give a similar counting result for the continued fractions semigroup of {\mathrm{SL}_{2}(\mathbf{Z})}, which arises in the study of Zaremba’s conjecture on continued fractions.

scholarly journals Formations, bihomorphisms and natural transformations

2000 ◽  
Vol 61 (2) ◽  
pp. 217-239
Author(s):  
Andrew Ensor
Keyword(s):  
General Properties ◽  
Implication Algebras ◽  
Uniform Congruence ◽  
Natural Transformations ◽  
Modular Varieties

Given a variety ν and ν-algebras A and B, an algebraic formationF: A ⇉ B is a ν-homomorphism FL R × A → B, for some ν-algebra R, and the resulting functions F (r,-): A → B for r ∈ R are termed formable. Firstly, as motivation for the study of algebraic formations, categorical formations and their relationship with natural transformations are explained. Then, formations and formable functions are described for some common varieties of algebras, including semilattices, lattices, groups, and implication algebras. Some of their general properties are investigated for congruence modular varieties, including the description of a uniform congruence which provides information on the structure of B.

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