For what kind of a subset its sums equal $\mathbb{R}^4$

Question

For short, suppose $a,b$ are real numbers. Let $A=\{(\cos(at), \cos(bt), \sin(at), \sin(bt))\mid t\in \mathbb{R}\}$.

Let $B=\sum A=\{\sum_{i=1}^n x_i\mid x_i\in A, n \geq 1\}$.

For what values $a,b$, $B$ equals $\mathbb{R}^4$?

In general, what conditions can we impose to a subset $A$ of $\mathbb{R}^n$, such that the sums of $A$ is the whole space?

Any references, suggestions are appreciated.

Thanks!

Out of the box I'd guess the following conditions: The origin must be in the convex hull of $A\setminus{0}$. And ${\sum_{i=1}^{\dim V}x_i\mid x_i\in A}$ must have an interior point. — Hagen von Eitzen, Jul 07 '13 at 08:31
It might be easier to deal With this as $\mathbb C^2$ instead... — Thomas Andrews, Jul 07 '13 at 08:51
For the general case, you clearly necessarily need it to be uncountable. — Thomas Andrews, Jul 07 '13 at 09:00

score 1 · Accepted Answer · answered Jul 07 '13 at 09:08

First notice that it is enough to prove that $B$ contains a neighbourhood of the origin. To prove that you consider the map: $$ \psi(t_1,\dots,t_n) = \sum_{i=1}^n \phi(t_i) $$ where $\phi(t)$ is the curve defining $A$.

First of all you want $A$ to contain the origin, so you have to solve $\psi(\bar t) = 0$ and see if you find conditions on your parameters $a$ and $b$.

Then you have some $\bar t_0$ such that $\psi(\bar t_0)=0$. So the map contains a neighbourhood of $0$ if the differential $D\psi(\bar t_0)$ has rank equal to $4$.

For what kind of a subset its sums equal $\mathbb{R}^4$

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