Bounds on total number of subspaces of finite field

Question

Are there asymptotic upper and lower bounds on the "total" number of subspaces of $\mathbf{F}^n_q$? $q$ is fixed and $n$ increases

The following two answers are helpful but dont answer the question.

How to count number of bases and subspaces of a given dimension in a vector space over a finite field?

https://mathoverflow.net/questions/89934/sum-of-gaussian-binomial-coefficients

Why wouldn't they answer the question? The first answer alone gives an exact value if you sum over all $k$, so that's both an upper bound and a lower bound! — Arnaud Mortier, Feb 28 '18 at 10:40
Yes but what is value of sum, in closed form, as function of n,q — LyxNewbie, Feb 28 '18 at 10:41
That way anyone can compute anything computable. Added asymptotic to question — LyxNewbie, Feb 28 '18 at 10:43
This is a pointless conversation. "better" is relative to your metric of goodness. Please answer question. Thanks — LyxNewbie, Feb 28 '18 at 10:46
Asymptotic in $q$, with $n$ fixed? Asymptotic in $n$, with $q$ fixed? Asymptotic in $r$, with $q=p^r$, $p$ and $n$ fixed? — Gerry Myerson, Feb 28 '18 at 11:03
Thanks for your answer. It gives tighter bounds on lower order terms, though only for q=2,3. I like my answer more since it is asymptotically tight for any $q$, with the right power of $n$. — LyxNewbie, Mar 04 '18 at 10:17
Well, I only gave the results for $q=2,3$, but have you followed up on the links I gave, to see whether they lead to more general results? — Gerry Myerson, Mar 05 '18 at 09:29
I looked at the links but none seem talk about general results. Most even dont talk about asymptotics — LyxNewbie, Mar 08 '18 at 05:41

Gerry Myerson · Answer 1 · 2018-03-01T02:02:11.473

When $q=2$, these numbers are tabulated at A006116 where many references and links are given. Maybe some of those lead to other values of $q$ and more general questions. It says the number is asymptotic to $c2^{n^2/4}$, where $c = {\rm EllipticTheta}[3,0,1/2] / {\rm QPochhammer}[1/2,1/2] = 7.3719688014613\dots$ if $n$ is even and $c = {\rm EllipticTheta}[2,0,1/2] / {\rm QPochhammer}[1/2,1/2] = 7.3719494907662\dots$ if $n$ is odd.

When $q=3$, the numbers are tabulated at A006117, and again there's an asymptotic formula of the form $c3^{n^2/4}$, where $c$ is given in terms of EllipticTheta and QPochhammer. I'm sure there's more where those came from.

score 0 · Answer 2 · answered Feb 28 '18 at 11:20

One simple way is to lower bound for $k=n/2$. Then for the specific term, denominator is at most $q^{n^2/4}$, and numerator is at least $(0.1q^n)^{n/2}$, which gives lower bound of $q^{\Omega(n^2)}$. Same can be done for upper bound. A better asymptotic bound would be nice.

Bounds on total number of subspaces of finite field

2 Answers2