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Abstract What proportion of integers$$n \leq N$$may be expressed as$$x^2 + dy^2$$for some$$d \leq \Delta $$, with$$x,y$$integers? Writing$$\Delta = (\log N)^{\log 2} 2^{\alpha \sqrt {\log \log N}}$$for some$$\alpha \in (-\infty , \infty )$$, we show that the answer is$$\Phi (\alpha ) + o(1)$$, where$$\Phi $$is the Gaussian distribution function$$\Phi (\alpha ) = \frac {1}{\sqrt {2\pi }} \int ^{\alpha }_{-\infty } e^{-x^2/2} dx$$. A consequence of this is a phase transition: Almost none of the integers$$n \leq N$$can be represented by$$x^2 + dy^2$$with$$d \leq (\log N)^{\log 2 - \varepsilon }$$, but almost all of them can be represented by$$x^2 + dy^2$$with$$d \leq (\log N)^{\log 2 + \varepsilon}\kern-1.5pt$$.
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Stable laws for random dynamical systems
Abstract In this paper, we consider random dynamical systems formed by concatenating maps acting on the unit interval$$[0,1]$$in an independent and identically distributed (i.i.d.) fashion. Considered as a stationary Markov process, the random dynamical system possesses a unique stationary measure$$\nu $$. We consider a class of non-square-integrable observables$$\phi $$, mostly of form$$\phi (x)=d(x,x_0)^{-{1}/{\alpha }}$$, where$$x_0$$is a non-recurrent point (in particular a non-periodic point) satisfying some other genericity conditions and, more generally, regularly varying observables with index$$\alpha \in (0,2)$$. The two types of maps we concatenate are a class of piecewise$$C^2$$expanding maps and a class of intermittent maps possessing an indifferent fixed point at the origin. Under conditions on the dynamics and$$\alpha $$, we establish Poisson limit laws, convergence of scaled Birkhoff sums to a stable limit law, and functional stable limit laws in both the annealed and quenched case. The scaling constants for the limit laws for almost every quenched realization are the same as those of the annealed case and determined by$$\nu $$. This is in contrast to the scalings in quenched central limit theorems where the centering constants depend in a critical way upon the realization and are not the same for almost every realization.
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- Award ID(s):
- 2009923
- PAR ID:
- 10543218
- Publisher / Repository:
- Ergodic Theory and Dynamical Systems
- Date Published:
- Journal Name:
- Ergodic Theory and Dynamical Systems
- ISSN:
- 0143-3857
- Page Range / eLocation ID:
- 1 to 50
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/etds.2024.5
