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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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Degrees of maps and multiscale geometry
Abstract We study the degree of an L-Lipschitz map between Riemannian manifolds, proving new upper bounds and constructing new examples. For instance, if $$X_k$$ is the connected sum of k copies of $$\mathbb CP^2$$for$$k \ge 4$$, then we prove that the maximum degree of an L-Lipschitz self-map of $$X_k$$ is between $$C_1 L^4 (\log L)^{-4}$$ and $$C_2 L^4 (\log L)^{-1/2}$$. More generally, we divide simply connected manifolds into three topological types with three different behaviors. Each type is defined by purely topological criteria. For scalable simply connected n-manifolds, the maximal degree is $$\sim L^n$$. For formal but nonscalable simply connectedn-manifolds, the maximal degree grows roughly like $$L^n (\log L)^{-\theta (1)}$$. And for nonformal simply connected n-manifolds, the maximal degree is bounded by $$L^\alpha $$ for some $$\alpha < n$$.
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- Award ID(s):
- 2204001
- PAR ID:
- 10562505
- Publisher / Repository:
- Cambridge University Press
- Date Published:
- Journal Name:
- Forum of Mathematics, Pi
- Volume:
- 12
- ISSN:
- 2050-5086
- Page Range / eLocation ID:
- e2
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/fmp.2023.33
