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Poisson distribution for gaps between sums of two squares and level spacings for toral point scatterers

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Tristan Freiberg, Pär Kurlberg, Lior Rosenzweig

Mathematical Physics · Submitted 4 Jan 2017

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Extending work of Connors--Keating, and Smilansky, we formulate an analog of the Hardy--Littlewood prime $k$-tuple conjecture for sums of two squares, and show that it implies that the spectral gaps, after removing degeneracies and rescaling, are Poisson distributed.

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arXiv ID1701.01157 (version 1)
Pith IDLNTVNACP, Pith's permanent record of this paper and its review
CitationLior Rosenzweig, P\"ar Kurlberg, Tristan Freiberg. "Poisson distribution for gaps between sums of two squares and level spacings for toral point scatterers." arXiv:1701.01157 (2017). https://pith.science/paper/LNTVNACP
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,667 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.23 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.17 · compares with prior results 0.77 · mentions code or data availability 0.03.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We investigate the level spacing distribution for the quantum spectrum of the square billiard. Extending work of Connors--Keating, and Smilansky, we formulate an analog of the Hardy--Littlewood prime $k$-tuple conjecture for sums of two squares, and show that it implies that the spectral gaps, after removing degeneracies and rescaling, are Poisson distributed. Consequently, by work of Rudnick and Uebersch\"ar, the level spacings of arithmetic toral point scatterers, in the weak coupling limit, are also Poisson distributed. We also give numerical evidence for the conjecture and its implications.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 We investigate the level spacing distribution for the quantum spectrum of the square billiard. contribution 0.66 0.79 0.03 asserted 0.60
s1 Extending work of Connors--Keating, and Smilansky, we formulate an analog of the Hardy--Littlewood prime $k$-tuple conjecture for sums of two squares, and show that it implies that the spectral gaps, after removing degeneracies and rescaling, are Poisson distributed. contribution 0.56 0.89 0.83 asserted 0.99
s2 Consequently, by work of Rudnick and Uebersch\"ar, the level spacings of arithmetic toral point scatterers, in the weak coupling limit, are also Poisson distributed. result 0.64 0.12 0.24 asserted 1.00
s3 We also give numerical evidence for the conjecture and its implications. contribution 0.60 0.92 0.05 asserted 0.89

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.92.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

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Original abstract

We investigate the level spacing distribution for the quantum spectrum of the square billiard. Extending work of Connors--Keating, and Smilansky, we formulate an analog of the Hardy--Littlewood prime $k$-tuple conjecture for sums of two squares, and show that it implies that the spectral gaps, after removing degeneracies and rescaling, are Poisson distributed. Consequently, by work of Rudnick and Uebersch\"ar, the level spacings of arithmetic toral point scatterers, in the weak coupling limit, are also Poisson distributed. We also give numerical evidence for the conjecture and its implications.

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