7 papers
The quality plane of -triples and the decomposition into gains
R. Laniewski, K. Müller
To a coprime triple we attach two coordinates, the quality and the logarithmic mass , whose ratio measures the balance of the triple and lies asymptotically betw…
Weak Hall's conjecture, approximation gains, and continued fractions
R. Laniewski, K. Müller
For an integer solution of with and , the integers , and form a triple in which one term is the sum of the other two. When $\gcd(x,y…
The Bombieri--van der Poorten Formula for Partial Quotients of Higher Degree Algebraic Irrationals
Karsten Müller
The fundamental relationship between the partial quotients of an algebraic irrational and its corresponding algebraic form was…
Gain Bounds for Diagonal Superelliptic Equations under the Strong ABC Conjecture
Karsten Müller
We establish a novel framework for bounding the adapted power gain and approximation gain of coprime integer solutions to the generalized diagonal superelliptic equatio…
From ABC to Effective Roth and Ridout Constants for Cubic Roots
Karsten Müller, Michael Taktikos
Enrico Bombieri showed conditionally (1994) that the ABC conjecture implies Roth's theorem, and Van Frankenhuysen (1999) later provided a complete proof. Building on Bombieri's and…
Do algebraic numbers follow Khinchin's Law?
Philipp Sibbertsen, Timm Lampert, Karsten Müller +1
The coefficients of the regular continued fraction for random numbers are distributed by the Gauss-Kuzmin distribution according to Khinchin's law. Their geometric mean converges t…
Roth's Theorem implies a Weakened Version of the ABC Conjecture for Special Cases
Philipp Sibbertsen, Timm Lampert, Karsten Müller +1
Enrico Bombieri proved that the ABC Conjecture implies Roth's theorem in 1994. This paper concerns the other direction. In making use of Bombieri's and Van der Poorten's explicit f…