23 research outputs found
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
On the interplay between Babai and Černý’s conjectures
Motivated by the Babai conjecture and the Černý conjecture, we study the reset thresholds of automata with the transition monoid equal to the full monoid of transformations of the state set. For automata with n states in this class, we prove that the reset thresholds are upperbounded by 2n2 -6n + 5 and can attain the value (Formula presented). In addition, we study diameters of the pair digraphs of permutation automata and construct n-state permutation automata with diameter (formula presented). © Springer International Publishing AG 2017
On the square-free sieve
We improve on the best available bounds for the square-free sieve and provide
a general framework for its applicability. The failure of the local-to-global
principle allows us to obtain results better than those reached by a classical
sieve-based approach. Techniques involving sphere-packing yield upper bounds on
the number of integer and rational points on curves of positive genus.Comment: 51 pages; typos fixe