160 research outputs found
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the first vertices has minimum degree .Comment: This paper has been withdrawn because we have found an easier proof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
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
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
Dense sets of integers with prescribed representation functions
Let A be a set of integers and let h \geq 2. For every integer n, let r_{A,
h}(n) denote the number of representations of n in the form n=a_1+...+a_h,
where a_1,...,a_h belong to the set A, and a_1\leq ... \leq a_h. The function
r_{A,h} from the integers Z to the nonnegative integers N_0 U {\infty} is
called the representation function of order h for the set A. We prove that
every function f from Z to N_0 U {\infty} satisfying liminf_{|n|->\infty} f
(n)\geq g is the representation function of order h for some sequence A of
integers, and that A can be constructed so that it increases "almost" as slowly
as any given B_h[g] sequence. In particular, for every epsilon >0 and g \geq
g(h,epsilon), we can construct a sequence A satisfying r_{A,h}=f and A(x)\gg
x^{(1/h)-epsilon}.Comment: 10 page
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