@article{BeckFiala,
 author = {Beck, J{\'o}zsef and Fiala, Tibor},
 title = {{``Integer-making'' theorems}},
 journal = {Discrete Applied Mathematics},
 volume = {3}, number = {1}, pages = {1--8}, year = {1981},
 note = {\href{https://doi.org/10.1016/0166-218X(81)90022-6}{doi:10.1016/0166-218X(81)90022-6}. The classical floating-variable proof is also given in Nikolov's Lecture 3.}
}
@article{Bergelson,
 author = {Bergelson, Vitaly and Ku{\l}aga-Przymus, Joanna and Lema{\'n}czyk, Mariusz and Richter, Florian K.},
 title = {Rationally almost periodic sequences, polynomial multiple recurrence and symbolic dynamics},
 journal = {Ergodic Theory and Dynamical Systems},
 volume = {39}, number = {9}, pages = {2332--2383}, year = {2019},
 note = {\href{https://doi.org/10.1017/etds.2017.130}{doi:10.1017/etds.2017.130}. Author version: \href{https://arxiv.org/abs/1611.08392v2}{arXiv:1611.08392v2}, Theorem 1.7.}
}
@article{BD,
 author = {de la Bret{\`e}che, R{\'e}gis and Drappeau, Sary},
 title = {Niveau de r{\'e}partition des polyn{\^o}mes quadratiques et crible majorant pour les entiers friables},
 journal = {Journal of the European Mathematical Society},
 volume = {22}, number = {5}, pages = {1577--1624}, year = {2020},
 note = {\href{https://doi.org/10.4171/JEMS/951}{doi:10.4171/JEMS/951}. Locators: Theorem 1.2 and Section 8.4 of \href{https://arxiv.org/abs/1703.03197v3}{arXiv:1703.03197v3}, 7 May 2019.}
}
@article{DI,
 author = {Deshouillers, Jean-Marc and Iwaniec, Henryk},
 title = {On the greatest prime factor of {$n^2+1$}},
 journal = {Annales de l'Institut Fourier},
 volume = {32}, number = {4}, pages = {1--11}, year = {1982},
 note = {Section 8. \url{https://www.numdam.org/item/AIF_1982__32_4_1_0/}.}
}
@article{GHR,
 author = {Galatolo, Stefano and Hoyrup, Mathieu and Rojas, Crist{\'o}bal},
 title = {A constructive {Borel--Cantelli} lemma. {Constructing} orbits with required statistical properties},
 journal = {Theoretical Computer Science},
 volume = {410}, number = {21--23}, pages = {2207--2222}, year = {2009},
 note = {\href{https://doi.org/10.1016/j.tcs.2009.02.010}{doi:10.1016/j.tcs.2009.02.010}. Author version: \href{https://arxiv.org/abs/0711.1478v2}{arXiv:0711.1478v2}. Context for effective statistical selection, not a black-box input.}
}
@misc{Guruswami6,
 author = {Guruswami, Venkatesan},
 title = {Introduction to Coding Theory. {Notes 6}: {Reed--Solomon}, {BCH}, {Reed--Muller}, and concatenated codes},
 year = {2010},
 howpublished = {Carnegie Mellon University, February 2010},
 note = {Scribes: Eric Blais and Venkat Guruswami. Sections 1 and 4. \url{https://www.cs.cmu.edu/~venkatg/teaching/codingtheory/notes/notes6.pdf}.}
}
@misc{Guruswami7,
 author = {Guruswami, Venkatesan},
 title = {Introduction to Coding Theory. {Notes 7}: {Justesen} codes, {Reed--Solomon} and concatenated codes decoding},
 year = {2010},
 howpublished = {Carnegie Mellon University, March 2010},
 note = {Scribes: Venkat Guruswami and Balakrishnan Narayanaswamy. Sections 3.3 and 5. \url{https://www.cs.cmu.edu/~venkatg/teaching/codingtheory/notes/notes7.pdf}.}
}
@article{Klurman,
 author = {Klurman, Oleksiy},
 title = {Correlations of multiplicative functions and applications},
 journal = {Compositio Mathematica},
 volume = {153}, number = {8}, pages = {1622--1657}, year = {2017},
 note = {\href{https://doi.org/10.1112/S0010437X17007163}{doi:10.1112/S0010437X17007163}. Locator for the large-prime argument: \href{https://arxiv.org/abs/1603.08453v1}{arXiv:1603.08453v1}, Section 2, proofs of Lemma 2.1 and Proposition 1.1.}
}
@misc{Nikolov,
 author = {Nikolov, Aleksandar},
 title = {{Prague Summer School: Discrepancy Theory. Lecture 3}},
 year = {2018},
 note = {Theorem 2, pages 1--2. \url{https://www.cs.toronto.edu/~anikolov/Prague18/Lectures/lect3.pdf}.}
}
@article{Pascadi,
 author = {Pascadi, Alexandru},
 title = {Large sieve inequalities for exceptional {Maass} forms and the greatest prime factor of {$n^2+1$}},
 journal = {Forum of Mathematics, Pi}, volume = {14}, pages = {e8}, year = {2026},
 note = {\href{https://doi.org/10.1017/fmp.2026.10025}{doi:10.1017/fmp.2026.10025}. Locators refer to \href{https://arxiv.org/abs/2404.04239v3}{arXiv:2404.04239v3}, 15 January 2026, Sections 6.1 and 6.3, equation (6.20).}
}
@misc{Raginsky,
 author = {Raginsky, Maxim},
 title = {Concentration inequalities},
 year = {2013},
 howpublished = {Lecture notes, 29 August 2013},
 note = {Theorem 2, page 6. \url{https://maxim.ece.illinois.edu/teaching/fall13/notes/concentration.pdf}.}
}
@inproceedings{Sukhavasi,
 author = {Sukhavasi, Ravi Teja and Hassibi, Babak},
 title = {Linear error correcting codes with anytime reliability},
 booktitle = {2011 IEEE International Symposium on Information Theory Proceedings},
 pages = {1748--1752}, year = {2011},
 note = {\href{https://doi.org/10.1109/ISIT.2011.6033847}{doi:10.1109/ISIT.2011.6033847}. Author record: \url{https://authors.library.caltech.edu/records/4jyv2-64h44}.}
}
@misc{DLMF,
 author = {{National Institute of Standards and Technology}},
 title = {{Digital Library of Mathematical Functions}, Section 27.11: Asymptotic Formulas: Partial Sums},
 howpublished = {\url{https://dlmf.nist.gov/27.11}},
 note = {Equation (27.11.11) and the fixed-progression prime number theorem. Accessed 27 September 2026.}
}

@incollection{EverestHarman,
 author = {Everest, Graham and Harman, Glyn},
 title = {On primitive divisors of {$n^2+b$}},
 editor = {McKee, James and Smyth, Chris},
 booktitle = {Number Theory and Polynomials},
 series = {London Mathematical Society Lecture Note Series},
 volume = {352}, pages = {142--154}, publisher = {Cambridge University Press}, year = {2008},
 note = {\href{https://doi.org/10.1017/CBO9780511721274.011}{doi:10.1017/CBO9780511721274.011}. Proposition 1.3; author preprint \href{https://arxiv.org/abs/math/0701234}{arXiv:math/0701234} (2007).}
}
@article{Bohus,
 author = {Bohus, G{\'e}za},
 title = {On the discrepancy of 3 permutations},
 journal = {Random Structures \& Algorithms},
 volume = {1}, number = {2}, pages = {215--220}, year = {1990},
 note = {\href{https://doi.org/10.1002/rsa.3240010208}{doi:10.1002/rsa.3240010208}.}
}
@misc{NewmanNikolov,
 author = {Newman, Alantha and Nikolov, Aleksandar},
 title = {A counterexample to {Beck's} conjecture on the discrepancy of three permutations},
 year = {2011},
 howpublished = {\href{https://arxiv.org/abs/1104.2922v1}{arXiv:1104.2922v1}, 14 April 2011}
}
@misc{GrimmeltMerikoski,
 author = {Grimmelt, Lasse and Merikoski, Jori},
 title = {On the greatest prime factor and uniform equidistribution of quadratic polynomials},
 year = {2025},
 howpublished = {\href{https://arxiv.org/abs/2505.00493v2}{arXiv:2505.00493v2}, 30 May 2025},
 note = {Theorem 1.1. Preprint cited for context, not as an input to the coding theorem.}
}
@misc{Li,
 author = {Li, Runbo},
 title = {On the largest prime factor of quadratic polynomials},
 year = {2025},
 howpublished = {\href{https://arxiv.org/abs/2406.07575v2}{arXiv:2406.07575v2}, 31 May 2025},
 note = {Theorem 1.1. This title and statement are those of version 2, not the different version 1 of 2024; context only.}
}
@article{ReedSolomon,
 author = {Reed, Irving S. and Solomon, Gustave},
 title = {Polynomial codes over certain finite fields},
 journal = {Journal of the Society for Industrial and Applied Mathematics},
 volume = {8}, number = {2}, pages = {300--304}, year = {1960},
 note = {\href{https://doi.org/10.1137/0108018}{doi:10.1137/0108018}.}
}
@article{Forney,
 author = {Forney, Jr., G. David},
 title = {Generalized minimum distance decoding},
 journal = {IEEE Transactions on Information Theory},
 volume = {12}, number = {2}, pages = {125--131}, year = {1966},
 note = {\href{https://doi.org/10.1109/TIT.1966.1053873}{doi:10.1109/TIT.1966.1053873}.}
}
@incollection{McDiarmid,
 author = {McDiarmid, Colin},
 title = {On the method of bounded differences},
 booktitle = {Surveys in Combinatorics, 1989},
 pages = {148--188}, publisher = {Cambridge University Press}, year = {1989},
 note = {\href{https://doi.org/10.1017/CBO9781107359949.008}{doi:10.1017/\allowbreak CBO9781107359949.008}.}
}
@article{FLR,
 author = {Frantzikinakis, Nikos and Lema{\'n}czyk, Mariusz and de la Rue, Thierry},
 title = {{Furstenberg} systems of pretentious and {MRT} multiplicative functions},
 journal = {Ergodic Theory and Dynamical Systems},
 volume = {45}, number = {9}, pages = {2765--2844}, year = {2025},
 note = {\href{https://doi.org/10.1017/etds.2024.140}{doi:10.1017/etds.2024.140}. Broader context; not a proof input here.}
}

@misc{PoulySelective,
 author = {Pouly, Brice},
 title = {Selective normality and algebraic rigidity of multiplicative traces},
 year = {2026},
 howpublished = {Zenodo},
 doi = {10.5281/zenodo.23318326},
 url = {https://doi.org/10.5281/zenodo.23318326},
 note = {\href{https://doi.org/10.5281/zenodo.23318326}{doi:10.5281/zenodo.23318326}. Section 5, especially Proposition 5.1.}
}
@misc{PoulyCorrelation,
 author = {Pouly, Brice},
 title = {Correlation laws and identifiability of multiplicative functions on quadratic traces},
 year = {2026},
 howpublished = {Zenodo},
 doi = {10.5281/zenodo.23318918},
 url = {https://doi.org/10.5281/zenodo.23318918},
 note = {\href{https://doi.org/10.5281/zenodo.23318918}{doi:10.5281/zenodo.23318918}. Section 7.}
}
@misc{PoulyReconstruction,
 author = {Pouly, Brice},
 title = {Arithmetic reconstruction and error recovery from three quadratic traces},
 year = {2026},
 howpublished = {Zenodo},
 doi = {10.5281/zenodo.23317860},
 url = {https://doi.org/10.5281/zenodo.23317860},
 note = {\href{https://doi.org/10.5281/zenodo.23317860}{doi:10.5281/zenodo.23317860}. Theorems 1.1--1.2.}
}