7 research outputs found
The ternary Goldbach problem
The ternary Goldbach conjecture, or three-primes problem, states that every
odd number greater than can be written as the sum of three primes. The
conjecture, posed in 1742, remained unsolved until now, in spite of great
progress in the twentieth century. In 2013 -- following a line of research
pioneered and developed by Hardy, Littlewood and Vinogradov, among others --
the author proved the conjecture.
In this, as in many other additive problems, what is at issue is really the
proper usage of the limited information we possess on the distribution of prime
numbers. The problem serves as a test and whetting-stone for techniques in
analysis and number theory -- and also as an incentive to think about the
relations between existing techniques with greater clarity.
We will go over the main ideas of the proof. The basic approach is based on
the circle method, the large sieve and exponential sums. For the purposes of
this overview, we will not need to work with explicit constants; however, we
will discuss what makes certain strategies and procedures not just effective,
but efficient, in the sense of leading to good constants. Still, our focus will
be on qualitative improvements.Comment: 29 pages. To be submitted to the Proceedings of the ICM 201
New dimensional estimates for subvarieties of linear algebraic groups
For every connected, almost simple linear algebraic group
over a large enough field , every subvariety
, and every finite generating set , we prove a
general {\em dimensional bound}, that is, a bound of the form with
depending only on . The dependence of on
(or rather on ) is doubly exponential, whereas (which is
independent of ) depends simply exponentially on .
Bounds of this form have proved useful in the study of growth in linear
algebraic groups since 2005 (Helfgott) and, before then, in the study of
subgroup structure (Larsen-Pink: a subgroup). In bounds for general and
available before our work, the dependence of and on was of
exponential-tower type.
We draw immediate consequences regarding diameter bounds for classical
Chevalley groups . (In a separate paper, we derive stronger
diameter bounds from stronger dimensional bounds we prove for specific families
of varieties .)Comment: 35 pages. Submitte
Growth in solvable subgroups of GL_r(Z/pZ)
Let and let be a subset of \GL_r(K) such that is
solvable. We reduce the study of the growth of $A$ under the group operation to
the nilpotent setting. Specifically we prove that either $A$ grows rapidly
(meaning $|A\cdot A\cdot A|\gg |A|^{1+\delta}$), or else there are groups $U_R$
and $S$, with $S/U_R$ nilpotent such that $A_k\cap S$ is large and
$U_R\subseteq A_k$, where $k$ is a bounded integer and $A_k = \{x_1 x_2...b x_k
: x_i \in A \cup A^{-1} \cup {1}}$. The implied constants depend only on the
rank $r$ of $\GL_r(K)$.
When combined with recent work by Pyber and Szab\'o, the main result of this
paper implies that it is possible to draw the same conclusions without
supposing that is solvable.Comment: 46 pages. This version includes revisions recommended by an anonymous
referee including, in particular, the statement of a new theorem, Theorem