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Circles passing through five or more integer points

Shaunna M. Plunkett-Levin (2013)

Acta Arithmetica

We find an improvement to Huxley and Konyagin’s current lower bound for the number of circles passing through five integer points. We conjecture that the improved lower bound is the asymptotic formula for the number of circles passing through five integer points. We generalise the result to circles passing through more than five integer points, giving the main theorem in terms of cyclic polygons with m integer point vertices. Theorem. Let m ≥ 4 be a fixed integer. Let W m ( R ) be the number of cyclic polygons...

Combinatoire du billard dans un polyèdre

Nicolas Bedaride (2006/2007)

Séminaire de théorie spectrale et géométrie

Ces notes ont pour but de rassembler les différents résultats de combinatoire des mots relatifs au billard polygonal et polyédral. On commence par rappeler quelques notions de combinatoire, puis on définit le billard, les notions utiles en dynamique et le codage de l’application. On énonce alors les résultats connus en dimension deux puis trois.

Concentration function of additive functions on shifted twin primes

Simon Wong (1998)

Acta Arithmetica

0. Introduction. The content of this paper is part of the author's Ph.D. thesis. The two new theorems in this paper provide upper bounds on the concentration function of additive functions evaluated on shifted γ-twin prime, where γ is any positive even integers. Both results are generalizations of theorems due to I. Z. Ruzsa, N. M. Timofeev, and P. D. T. A. Elliott.

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