A note on the pair correlation of Farey fractions
The pair correlations of Farey fractions with denominators $q$ satisfying $(q,m)=1$, respectively $q\equiv b \pmod{m}$ with $(b,m)=1$, are shown to exist and are explicitly computed.
math.NT↗
arXiv subjects
Maria Siskaki
Publications and source records attributed to Maria Siskaki.
On the Gauss-Kuzmin-L\'evy problem for nearest integer continued fractions
This note provides an effective bound in the Gauss-Kuzmin-L\'evy problem for some Gauss type shifts associated with nearest integer continued fractions, acting on the interval $I_0=[0,\frac{1}{2}]$ or $I_0=[-\frac{1}{2},\frac{1}{2}]$. We prove asymptotic formulas $\lambda (T^{-n}I) =\mu(I)(\vert I_0 \vert +O(q^n))$ for such transformations $T$, where $\lambda$ is the Lebesgue measure on $\mathbb R$, $\mu$ the normalized $T$-invariant Lebesgue absolutely continuous measure, $I$ subinterval in $I_0$, and $q=0.288$ is smaller than the Wirsing constant $q_W=0.3036\ldots$
math.NT↗
Distribution of the reduced quadratic irrationals arising from the odd continued fraction expansion
This paper investigates the quadratic irrationals that arise as periodic points of the Gauss type shift associated to the odd continued fraction expansion. It is shown that these numbers, which we call O-reduced, when ordered by the length of the associated closed primitive geodesic on some modular surface $Γ\backslash \mathbb{H}$, are equidistributed with respect to the Lebesgue absolutely continuous invariant probability measure of the Odd Gauss shift.
math.NT↗
Distribution of periodic points of certain Gauss shifts with infinite invariant measure
This paper investigates the periodic points of the Gauss type shifts associated to the even continued fraction (Schweiger) and to the backward continued fraction (Rényi). We show that they coincide exactly with two sets of quadratic irrationals that we call $E$-reduced, and respectively $B$-reduced. We prove that these numbers are equidistributed with respect to the (infinite) Lebesgue absolutely continuous invariant measures of the corresponding Gauss shift.
math.DS↗
Boundedness of derivatives and anti-derivatives of holomorphic functions as a rare phenomenon
In this article we prove a general result which in particular suggests that, on a simply connected domain in C, all the derivatives and anti-derivatives of the generic holomorphic function are unbounded. A similar result holds for the operator of partial sums of the Taylor expansion with center z at 0, seen as functions of the center z. We also discuss a universality result of these operators.
math.CV↗