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W. GEORG NOWAK, Universität für Bodenkultur, A-1180 Vienna, Austria |
Large convex domains sometimes contain more lattice points
than we would expect |
Let as usual r(n) denote the number of ways to write
as
a sum of two squares. Then the quantity
(the ``lattice rest'' of an origin-centered circle of radius
) is
well-known to satisfy
(Cramér, Landau),
(Hardy), and
(Corrádi & Kátai). Compared to (2), (3) is not only weaker but
also incapable of generalisation to more general domains, its proof
being based on rather special ``arithmetic'' arguments.
The present talk addresses the corresponding problem for cubes: Let
and denote by P_{3}(
t)
the error term in the asymptotic formula for
. Combining classic analytic number
theory with some profound algebra and a very recent deep result of
Heath-Brown [1], the authors [2] where able to show that
This is based on the fact that the set
contains a rather ``large'' set of numbers which are linearly
independent over the rationals.
The corresponding general problem for r_{k}(n), k>3, remains open at
the present state-of-art.
References
1. D. R. Heath-Brown, The density of rational points on
cubic
surfaces. Acta Arith. 79(1997), 17-30.
2. M. Kühleitner, W. G. Nowak, J. Schoißengeier and
T. Wooley,
On sums of two cubes: An -estimate for the error term.
Acta
Arith. 85(1998), 179-195.
Next: Yannis Petridis - Zeros
Up: Number Theory / Théorie
Previous: Kumar Murty - Zeros