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Harald Andres Helfgott

  1. Growth of small generating sets in SL_n(Z/pZ).

    Втр, 02/09/2010 - 16:01 — SomNambul
    Authors: Harald Andres Helfgott, Nick Gill
    Subjects: Group Theory
    Abstract

    Let G=SL_n. Let K=Z/pZ, p a prime. Let A\subset G(K) generate G(K). Suppose
    that |A|<p^{n+1-\delta}, delta>0. Then

    |A A A|>>|A|^{1+\epsilon}, where epsilon>0 and the implied constant depend
    only on n and delta.

  2. An explicit incidence theorem in F_p.

    Ср, 01/13/2010 - 16:01 — SomNambul
    Authors: Harald Andres Helfgott, Misha Rudnev
    Subjects: Combinatorics
    Abstract

    Let $P = A\times A \subset \mathbb{F}_p \times \mathbb{F}_p$, $p$ a prime.
    Assume that $P= A\times A$ has $n$ elements, $n<p$. See $P$ as a set of points
    in the plane over $\mathbb{F}_p$. We show that the pairs of points in $P$
    determine $\geq c n^{1 + {1/232}}$ lines, where $c$ is an absolute constant.

    We derive from this an incidence theorem: the number of incidences between a
    set of $n$ points and a set of $n$ lines in the projective plane over $\F_p$
    ($n<\sqrt{p}$) is bounded by $C n^{{3/2}-{1/9278}}$, where $C$ is an absolute
    constant.

  3. Improving Roth's theorem in the primes.

    Чт, 12/10/2009 - 16:01 — SomNambul
    Authors: Harald Andres Helfgott, Anne de Roton
    Subjects: Number Theory
    Abstract

    Let A be a subset of the primes. Let \delta_P(N) = \frac{|\{n\in A: n\leq
    N\}|}{|\{\text{$n$ prime}: n\leq N\}|}. We prove that, if \delta_P(N)\geq C
    \frac{\log \log \log N}{(\log \log N)^{1/3}} for N\geq N_0, where C and N_0 are
    absolute constants, then A\cap [1,N] contains a non-trivial three-term
    arithmetic progression.

    This improves on B. Green's result, which needs \delta_P(N) \geq C'
    \sqrt{\frac{\log \log \log \log \log N}{\log \log \log \log N}}.

  4. Azar y Aritmetica.

    Пнд, 09/07/2009 - 10:33 — SomNambul
    Authors: Harald Andres Helfgott
    Subjects: Probability
    Abstract

    Let omega(n) be the number of prime divisors of an integer n. Let n be an
    integer taken at random between 1 and N. What can be said about the value then
    taken by omega(n)? What is its expected value? What is its distribution in the
    limit? What is the probability that omega(n) will deviate greatly from its
    expected value?

  5. Azar y Aritmetica.

    Пнд, 09/07/2009 - 10:33 — SomNambul
    Authors: Harald Andres Helfgott
    Subjects: Probability
    Abstract

    Let omega(n) be the number of prime divisors of an integer n. Let n be an
    integer taken at random between 1 and N. What can be said about the value then
    taken by omega(n)? What is its expected value? What is its distribution in the
    limit? What is the probability that omega(n) will deviate greatly from its
    expected value?

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