Steven Senger: July 2012

Tuesday, July 31, 2012

Firestone's Union Jack IPA (A-)

Okay, so I really don't know what I'm talking about, but here are my impressions. I really like this beer. It gets a solid A-. Firestone is out of Paso Robles, CA. This is the first beer of theirs that I've tasted, and I am excited to try more.

The front end is malty and almost sweet. It is vaguely reminiscent of a barley wine, but still quite light, and not syrupy. The middle shows the hops a bit, but with very little harshness. The bitterness was pretty spot on where I like it-- just enough to show you that this is indeed an IPA, but not a showy bitterness like Stone's Arrogant Bastard. It rolls into the finish smoothly. I know that this is a cheesy thing to say, but it is really well balanced. Also, it clocks in at 7.5% abv.

Monday, July 30, 2012

Rich lines or planes in vector spaces over finite fields

This is just a little note. I visited my advisor, Alex Iosevich a few weeks ago. He has some undergraduates doing research, and they remarked that they could prove some result if they could show that a large subset of the three-dimensional vector space over a finite field must have some plane with a lot of points. It turns out that this statement, as well as a related statement in a two-dimensional vector space is not true in general. Here are the precise statements.

$\dpi{100} $There exists set $E\subset \mathbb{F}_q^3$ such that $|E|\gtrsim q^\frac{3}{2}$, and no plane contains $\gtrsim q$ points of the set $E$.$
Flat things and curved things share very little in common. So we consider a “sphere” in a vector space over a finite field. $\dpi{100} $ A ``sphere'' in of radius $r$ in $\mathbb{F}_q^d$ will be defined to be the set of points satisfying the equation $x_1^2+x_2^2+\dots +x_d^2=r.$ Note that we do not take a square root of the sum.$
The trick is to consider a special type of field, and consider a unit sphere in that field. $\dpi{100} $ Let $q=p^4$. Now, consider the sphere of radius one centered at the origin in the 3-dimensional vector space over the finite field of order $p^3$. The sphere has about $p^6= q^{3/2}$ points on it, and no plane will have more than about $p^3=q^{3/4}$ points.$

The statement in two dimensions is similar.
$\dpi{100} $There exists a set $E\subset\mathbb{F}_q^2$ such that $|E|\gtrsim q^\frac{3}{2}$, and no line contains more than $\gtrsim q^\frac{1}{2}$ points of the set $E$.$
Again, the proof involves curved things, but here, we use a slightly different approach.
$\dpi{100} $Let our set be the union of $q^{1/2}$ circles of different radii, centered at the origin. No line can hit any circle more than twice, so each line has fewer than $2q^{1/2}$ points from our set. Each circle has about $q$ points on it, for a grand total of $q^{3/2}$ points in our set.$

Clearly, there is plenty of room for more investigation here. I wonder how these statements would change if we were curious about the population of spheres and circles as opposed to lines and planes. It should be no surprise to anyone that knows me that this begins to look like the Erdős single distance problem.

Math notation

I tend to use the same notation pretty consistently, but it might not be standard for everyone. Here is a list of some things I use commonly. $\dpi{100} $If $E$ is some set, I often use $|E|$ to mean the size of the set. If $A$ is a subset of some ring, I typically denote its sum set by $A+A=\{a+b:a,b\in A \}$, and its product set by $AA=\{ab:a,b\in A \}$.$ $\dpi{100} $If $X(n)$ and $Y(n)$ are two functions that change as some parameter $n$ grows large, then we write $X(n)\lesssim Y(n)$ if there exist some positive constants, $C$ and $N$, independent of $n$, such that for all $n\geq N$, $X(n)\leq CY(n).$ If $X(n)\lesssim Y(n)$ and $Y(n)\lesssim X(n)$, we write $X(n)\approx Y(n)$. To also bury logarithmic factors, we write $X(n)\lessapprox Y(n)$ when for every $\epsilon>0$, there exist some positive constants, $C_\epsilon$ and $N_\epsilon$, independent of $n$, such that for all $n\geq N_\epsilon, X(n)\leq C_\epsilon Y(n).$$

Sunday, July 29, 2012

Something else to abandon

Hey!

I've been meaning to get back on the blog bandwagon for a while now. I have a few things about which I'd like to chatter for myself and a handful of others. For better info about for my mathematics, or some links to my music and rock climbing stuff, you're probably better off going to my homepage.