References
This page gives the immediate literature context for the current theorem package. It is not meant to replace the working bibliography in book/references.bib, but it does collect the papers most relevant to first contact.
Entry Point
- Igor E. Shparlinski, Modular hyperbolas, Japanese Journal of Mathematics 7 (2012).
This is the cleanest survey entry point for the surrounding literature and the natural first reference for explaining how the present project relates to earlier work on congruence-defined point sets.
Closest Geometric Neighbors
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Sergei V. Konyagin and Igor E. Shparlinski, On the Convex Hull of the Points on Modular Hyperbolas (arXiv, 2010). Relevance: convex-hull questions for modular hyperbolas are already part of the literature; this project differs by keeping the whole residue family and the area data in view.
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Kevin Ford, Mizan R. Khan, and Igor E. Shparlinski, Geometric properties of points on modular hyperbolas, Proceedings of the American Mathematical Society 138 (2010). Relevance: geometric behavior of points on modular hyperbolas, especially in the unit case.
Nearby Arithmetic-Geometric Context
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Javier Cilleruelo and Moubariz Z. Garaev, Concentration of Points on Two and Three Dimensional Modular Hyperbolas and Applications, Geometric and Functional Analysis 21 (2011).
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Igor E. Shparlinski and Arne Winterhof, On the number of distances between the coordinates of points on modular hyperbolas, Journal of Number Theory 128 (2008).
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Igor E. Shparlinski, On the distribution of points on multidimensional modular hyperbolas, Proceedings of the Japan Academy, Series A, Mathematical Sciences 83 (2007).
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Igor E. Shparlinski and Jose Felipe Voloch, Visible points on curves over finite fields, Bulletin of the Polish Academy of Sciences: Mathematics 55 (2007).
How The Present Project Differs
The current project is not only about one congruence class $$ xy \equiv a \pmod N $$ at a time.
Its distinctive emphasis is:
- the full residue family $\{A_{N,0},\ldots,A_{N,N-1}\}$
- the convex-hull areas $S(N,a)$ and the total $S(N)$
- the visible role of the zero class and zero-divisor geometry
- exact boundary-layer models for the full multiplication-table picture
For the longer project-facing discussion, see /wiki/book/04_modular_hyperbolas_in_context.