Explicit irrationality measures for continued fractions

AbstractLet τ=[a0;a1,a2,…], a0∈N, an∈Z+, n∈Z+, be a simple continued fraction determined by an infinite integer sequence (an). We are interested in finding an effective irrationality measure as explicit as possible for the irrational number τ. In particular, our interest is focused on sequences (an) with an upper bound at most (ank), where a>1 and k>0. In addition to our main target, arithmetic of continued fractions, we shall pay special attention to studying the nature of the inverse function z(y) of y(z)=zlogz

Insights into bear evolution from a Pleistocene polar bear genome

The polar bear (Ursus maritimus) has become a symbol of the threat to biodiversity from climate change. Understanding polar bear evolutionary history may provide insights into apex carnivore responses and prospects during periods of extreme environmental perturbations. In recent years, genomic studies have examined bear speciation and population history, including evidence for ancient admixture between polar bears and brown bears (Ursus arctos). Here, we extend our earlier studies of a 130,000- to 115,000-y-old polar bear from the Svalbard Archipelago using a 10x coverage genome sequence and 10 new genomes of polar and brown bears from contemporary zones of overlap in northern Alaska. We demonstrate a dramatic decline in effective population size for this ancient polar bear's lineage, followed by a modest increase just before its demise. A slightly higher genetic diversity in the ancient polar bear suggests a severe genetic erosion over a prolonged bottleneck in modern polar bears. Statistical fitting of data to alternative admixture graph scenarios favors at least one ancient introgression event from brown bears into the ancestor of polar bears, possibly dating back over 150,000 y. Gene flow was likely bidirectional, but allelic transfer from brown into polar bear is the strongest detected signal, which contrasts with other published work. These findings may have implications for our understanding of climate change impacts: Polar bears, a specialist Arctic lineage, may not only have undergone severe genetic bottlenecks but also been the recipient of generalist, boreal genetic variants from brown bears during critical phases of Northern Hemisphere glacial oscillations.Peer reviewe

Constructive Diophantine approximation in generalized continued fraction Cantor sets

Abstract We study which asymptotic irrationality exponents are possible for numbers in generalized continued fraction Cantor sets $$E_{\mathcal B}^{\mathcal A} = \Biggl\{ \frac{a_1}{b_1+\dfrac{a_2}{b_2+\cdots}}\colon a_n \in {\mathcal A},\, b_n \in {\mathcal B} \text{ for all } n \Biggr\},$$ where ${\mathcal A}$ and ${\mathcal B}$ are some given finite sets of positive integers. We give sufficient conditions for $E^{\mathcal A}_{\mathcal B}$ to contain numbers for any possible asymptotic irrationality exponent and show that sets with this property can have arbitrarily small Hausdorff dimension. We also show that it is possible for $E^{\mathcal A}_{\mathcal B}$ to contain very well approximable numbers even though the asymptotic irrationality exponents of the numbers in $E^{\mathcal A}_{\mathcal B}$ are bounded

On irrationality exponents of generalized continued fractions

Abstract We study how the asymptotic irrationality exponent of a given generalized continued fraction behaves as a function of growth properties of partial coefficient sequences ( ) and ( )

Rational approximations of the exponential function at rational points

We give explicit and asymptotic lower bounds for the quantity $|e^{s/t} - M/N|$ by studying a generalized continued fraction expansion of $e^{s/t}$. In cases $|s| \ge 3$ we improve existing results by extracting a large common factor from the numerators and the denominators of the convergents of that generalized continued fraction