88 research outputs found
Contractions and expansion
Let A be a finite set of reals and let K >= 1 be a real number. Suppose that
for each a in A we are given an injective map f_a : A -> R which fixes a and
contracts other points towards it in the sense that |a - f_a(x)| <= |a - x|/K
for all x in A, and such that f_a(x) always lies between a and x. Then the
union of the f_a(A) has cardinality >= K|A|/10 - O_K(1). An immediate
consequence of this is the estimate |A + K.A| >= K|A|/10 - O_K(1), which is a
slightly weakened version of a result of Bukh.Comment: 6 pages, submitted to special volume of EJC in honour of Yahya
Hamidoun
Approximate groups, I: the torsion-free nilpotent case
We describe the structure of ``K-approximate subgroups'' of torsion-free
nilpotent groups, paying particular attention to Lie groups.
Three other works, by Fisher-Katz-Peng, Sanders and Tao, have appeared which
independently address related issues. We comment briefly on some of the
connections between these papers.Comment: 23 page
Projective geometries arising from Elekes-Szab\'o problems
We generalise the Elekes-Szab\'o theorem to arbitrary arity and dimension and
characterise the complex algebraic varieties without power saving. The
characterisation involves certain algebraic subgroups of commutative algebraic
groups endowed with an extra structure arising from a skew field of
endomorphisms. We also extend the Erd\H{o}s-Szemer\'edi sum-product phenomenon
to elliptic curves. Our approach is based on Hrushovski's framework of
pseudo-finite dimensions and the abelian group configuration theorem.Comment: 48 pages. Minor improvements in presentation. To appear in ASEN
Approximate groups, II: the solvable linear case
We describe the structure of "K-approximate subgroups'' of solvable subgroups
of GL_n(C), showing that they have a large nilpotent piece. By combining this
with the main result of our recent paper on approximate subgroups of
torsion-free nilpotent groups, we show that such approximate subgroups are
efficiently controlled by nilpotent progressions.Comment: 10 page
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