165 research outputs found
Lattice points on circles, squares in arithmetic progressions and sumsets of squares
Rudin conjectured that there are never more than c N^(1/2) squares in an
arithmetic progression of length N. Motivated by this surprisingly difficult
problem we formulate more than twenty conjectures in harmonic analysis,
analytic number theory, arithmetic geometry, discrete geometry and additive
combinatorics (some old and some new) which each, if true, would shed light on
Rudin's conjecture.Comment: 21 pages, preliminary version. Comments welcom
Combinatorial problems in finite fields and Sidon sets
We use Sidon sets to present an elementary method to study some combinatorial
problems in finite fields, such as sum product estimates, solubility of some
equations and distribution of sequences in small intervals. We obtain classic
and more recent results avoiding the use of exponential sums, the usual tool to
deal with these problems.Comment: 13 page
The least common multiple of a quadratic sequence
For any irreducible quadratic polynomial f(x) in Z[x] we obtain the estimate
log l.c.m.(f(1),...,f(n))= n log n + Bn + o(n) where B is a constant depending
on f.Comment: 26 page
Infinite graphs that do not contain cycles of length four
We construct a countable infinite graph G that does not contain cycles of
length four having the property that the sequence of graphs induced by
the first vertices has minimum degree .Comment: This paper has been withdrawn because we have found an easier proof
of the resul
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