arXiv subjects

Harald A. Helfgott

Publications and source records attributed to Harald A. Helfgott.

6 recordsLinked to original sources

arXiv2022

Explicit $L^2$ bounds for the Riemann $ζ$ function

Explicit bounds on the tails of the zeta function $ζ$ are needed for applications, notably for integrals involving $ζ$ on vertical lines or other paths going to infinity. Here we bound weighted $L^2$ norms of tails of $ζ$. Two approaches are followed, each giving the better result on a different range. The first one is inspired by the proof of the standard mean value theorem for Dirichlet polynomials. The second approach, superior for large $T$, is based on classical lines, starting with an approximation to $ζ$ via Euler-Maclaurin. Both bounds give main terms of the correct order for $0<σ\leq 1$ and are strong enough to be of practical use for the rigorous computation of improper integrals. We also present bounds for the $L^{2}$ norm of $ζ$ in $[1,T]$ for $0\leqσ\leq 1$.

math.NT↗

arXiv2022

Summing $μ(n)$: a faster elementary algorithm

We present a new elementary algorithm that takes \[ \mathrm{time} \ \ O_ε\left(x^{\frac{3}{5}} (\log x)^{\frac{3}{5}+ε} \right) \ \ \mathrm{and}\ \ \mathrm{space} \ \ O\left(x^{\frac{3}{10}} (\log x)^{\frac{13}{10}} \right)\] for computing $M(x) = \sum_{n \leq x} μ(n),$ where $μ(n)$ is the Möbius function. This is the first improvement in the exponent of $x$ for an elementary algorithm since 1985. We also show that it is possible to reduce space consumption to $O(x^{1/5} (\log x)^{5/3})$ by the use of (Helfgott, 2020; arxiv.org:1712.09130), at the cost of letting time rise to the order of $x^{3/5} (\log x)$.

math.NT↗

arXiv2018

Growth in linear algebraic groups and permutation groups: towards a unified perspective

By now, we have a product theorem in every finite simple group $G$ of Lie type, with the strength of the bound depending only in the rank of $G$. Such theorems have numerous consequences: bounds on the diameters of Cayley graphs, spectral gaps, and so forth. For the alternating group Alt_n, we have a quasipolylogarithmic diameter bound (Helfgott-Seress 2014), but it does not rest on a product theorem. We shall revisit the proof of the bound for Alt_n, bringing it closer to the proof for linear algebraic groups, and making some common themes clearer. As a result, we will show how to prove a product theorem for Alt_n -- not of full strength, as that would be impossible, but strong enough to imply the diameter bound.

math.GR↗

arXiv2017

Soficity, short cycles and the Higman group

This is a paper with two aims. First, we show that the map from $\mathbb{Z}/p\mathbb{Z}$ to itself defined by exponentiation $x\to m^x$ has few $3$-cycles -- that is to say, the number of cycles of length three is $o(p)$. This improves on previous bounds. Our second objective is to contribute to an ongoing discussion on how to find a non-sofic group. In particular, we show that, if the Higman group were sofic, there would be a map from $\mathbb{Z}/p\mathbb{Z}$ to itself, locally like an exponential map, yet satisfying a recurrence property.

math.GR↗

arXiv2014

Random generators of the symmetric group: diameter, mixing time and spectral gap

Let $g$, $h$ be a random pair of generators of $G=Sym(n)$ or $G=Alt(n)$. We show that, with probability tending to $1$ as $n\to \infty$, (a) the diameter of $G$ with respect to $S = \{g,h,g^{-1},h^{-1}\}$ is at most $O(n^2 (\log n)^c)$, and (b) the mixing time of $G$ with respect to $S$ is at most $O(n^3 (\log n)^c)$. (Both $c$ and the implied constants are absolute.) These bounds are far lower than the strongest worst-case bounds known (in Helfgott--Seress, 2013); they roughly match the worst known examples. We also give an improved, though still non-constant, bound on the spectral gap. Our results rest on a combination of the algorithm in (Babai--Beals--Seress, 2004) and the fact that the action of a pair of random permutations is almost certain to act as an expander on $\ell$-tuples, where $\ell$ is an arbitrary constant (Friedman et al., 1998).

math.GR↗

arXiv2013

On the diameter of permutation groups

Given a finite group $G$ and a set $A$ of generators, the diameter diam$(Γ(G,A))$ of the Cayley graph $Γ(G,A)$ is the smallest $\ell$ such that every element of $G$ can be expressed as a word of length at most $\ell$ in $A \cup A^{-1}$. We are concerned with bounding diam(G):= $\max_A$ diam$(Γ(G,A))$. It has long been conjectured that the diameter of the symmetric group of degree $n$ is polynomially bounded in $n$, but the best previously known upper bound was exponential in $\sqrt{n \log n}$. We give a quasipolynomial upper bound, namely, \[\text{diam}(G) = \exp(O((\log n)^4 \log\log n)) = \exp((\log \log |G|)^{O(1)})\] for G = Sym(n) or G = \Alt(n), where the implied constants are absolute. This addresses a key open case of Babai's conjecture on diameters of simple groups. By standard results, our bound also implies a quasipolynomial upper bound on the diameter of all transitive permutation groups of degree $n$.

math.GR↗