Timeline for Show that $\mathrm{SL}_2(\mathbb{F}_p)$ is quasi-random
Current License: CC BY-SA 4.0
10 events
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| Feb 25, 2022 at 15:41 | comment | added | David E Speyer | Specifically, see Theorem 3.3 in Gowers. If $A$, $B$ and $C$ are subsets of $G$ then, if multiplication were a random map $G \times G \to G$, you would predict that the number of solutions to $ab=c$ with $a \in A$, $b \in B$, $c \in C$, would be about $|A| |B| |C|/|G|$. In a $D$-quasirandom group, if $|A| |B| |C| > |G|^3/(\epsilon^2 D)$, then the number of such solutions is at least $(1- \epsilon) |A| |B| |C|/|G|$. So, when $D$ is large, the hypothesis is easier to satisfy. | |
| Feb 25, 2022 at 15:29 | history | edited | YCor | CC BY-SA 4.0 |
formatting, added tag
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| Feb 25, 2022 at 15:10 | history | edited | coudy | CC BY-SA 4.0 |
correct spelling
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| May 26, 2016 at 10:16 | history | edited | Stefan Kohl♦ | CC BY-SA 3.0 |
Fixed a typo.
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| May 26, 2016 at 2:21 | comment | added | Yemon Choi | I don't have the background/intuition to answer that question at a conceptual level, but Gowers's paper does briefly review known notions of "quasi-random graph", and I think it is that definition which is supposed to be relevant to behvaiour of ${\rm SL}(2,p)$ | |
| May 25, 2016 at 19:50 | comment | added | john mangual | What's so random about $SL(2, \mathbb{F}_p)$? | |
| May 25, 2016 at 19:05 | comment | added | Geoff Robinson | @YemonChoi : Oops sorry, I see we are referring to the same paper! | |
| May 25, 2016 at 19:04 | comment | added | Geoff Robinson | I thought that the terminology originated with Gowers, and the notion was introduced to produce large product-free sets in non-Abelian groups. Gowers has a paper titled "Quasirandom groups" ( available online) in which he considers ${\rm PSL}(2,q)$ and uses the fact that it has no non-trivial irreducible character of degree less than $\frac{q-1}{2}$. | |
| May 25, 2016 at 18:49 | comment | added | Yemon Choi | I think the terminology either originates in, or was inspired by, this paper arxiv.org/abs/0710.3877 | |
| May 25, 2016 at 18:42 | history | asked | john mangual | CC BY-SA 3.0 |