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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...

Crible et 3-rang des corps quadratiques

Karim Belabas (1996)

Annales de l'institut Fourier

Considérons le cardinal h 3 * ( Δ ) de l’ensemble des racines cubiques de l’unité dans le groupe des classes de ( Δ ) , où Δ est un discriminant fondamental. Un résultat de Davenport et Heilbronn calcule la valeur moyenne de ces nombres quand Δ varie. On obtient ici géométriquement une borne explicite pour le reste, avec la possibilité supplémentaire de restreindre les Δ à des progressions arithmétiques. Des techniques de crible permettent alors d’évaluer la 3-partie des ( ± P k ) , où P k est pseudo-premier d’ordre k . On...

Critical Dimensions for counting Lattice Points in Euclidean Annuli

L. Parnovski, N. Sidorova (2010)

Mathematical Modelling of Natural Phenomena

We study the number of lattice points in ℝd, d ≥ 2, lying inside an annulus as a function of the centre of the annulus. The average number of lattice points there equals the volume of the annulus, and we study the L1 and L2 norms of the remainder. We say that a dimension is critical, if these norms do not have upper and lower bounds of the same order as the radius goes to infinity. In [Duke Math. J., 107 (2001), No. 2, 209–238], it was proved that...

Distribution of lattice points on hyperbolic surfaces

Vsevolod F. Lev (1996)

Acta Arithmetica

Let two lattices Λ ' , Λ ' ' s have the same number of points on each hyperbolic surface | x . . . x s | = C . We investigate the case when Λ’, Λ” are sublattices of s of the same prime index and show that then Λ’ and Λ” must coincide up to renumbering the coordinate axes and changing their directions.

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