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The spectral decomposition of $|\theta|^2$

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Paul D. Nelson

Number Theory · Submitted 11 Jan 2016

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

The authors' main claim We establish a spectral decomposition and surprisingly strong asymptotic formulas for $\langle |\theta|^2, \varphi \rangle$ as $\varphi$ traverses a sequence of Hecke-translates of a nice enough fixed function.

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arXiv ID1601.02529 (version 3)
Pith IDAZIFH3GJ, Pith's permanent record of this paper and its review
CitationPaul D. Nelson. "The spectral decomposition of $|\theta|^2$." arXiv:1601.02529 (2016). https://pith.science/paper/AZIFH3GJ
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,563 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.91 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.18 · compares with prior results 0.09 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; .

Let $\theta$ be an elementary theta function, such as the classical Jacobi theta function. We establish a spectral decomposition and surprisingly strong asymptotic formulas for $\langle |\theta|^2, \varphi \rangle$ as $\varphi$ traverses a sequence of Hecke-translates of a nice enough fixed function. The subtlety is that typically $|\theta|^2 \notin L^2$. Applications to the subconvexity, quantum variance and $4$-norm problems are indicated.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Let $\theta$ be an elementary theta function, such as the classical Jacobi theta function. background 0.99 0.05 0.02 asserted 0.86
s1 We establish a spectral decomposition and surprisingly strong asymptotic formulas for $\langle |\theta|^2, \varphi \rangle$ as $\varphi$ traverses a sequence of Hecke-translates of a nice enough fixed function. contribution 0.54 0.96 0.90 asserted 1.00
s2 The subtlety is that typically $|\theta|^2 \notin L^2$. background 0.45 0.33 0.05 asserted 0.98
s3 Applications to the subconvexity, quantum variance and $4$-norm problems are indicated. interpretation 0.51 0.41 0.04 hedged possibility 1.00

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

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  • P abstract
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  • C citations
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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

The paper
Original abstract

Let $\theta$ be an elementary theta function, such as the classical Jacobi theta function. We establish a spectral decomposition and surprisingly strong asymptotic formulas for $\langle |\theta|^2, \varphi \rangle$ as $\varphi$ traverses a sequence of Hecke-translates of a nice enough fixed function. The subtlety is that typically $|\theta|^2 \notin L^2$. Applications to the subconvexity, quantum variance and $4$-norm problems are indicated.

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Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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