I have been working on the following problem for a while, but seem to be at an impasse. My limited knowledge of additive combinatorics does not help. Suppose we have two positive real numbers $\alpha$ and $\beta$. We call $$S = \{m\alpha+n\beta:m,n\in\mathbb{N}\}$$ We are interested in growth rate of $$|\Gamma(n)=S\cap [0,n]|$$ Naturally, to avoid triviality, we assume $\alpha$ and $\beta$ are lineary intependent over $\mathbb{Z}$. E.g. (and I would be quite pleased with an awnser to this particular case) $\alpha = \sqrt{2}$ and $\beta = \sqrt{3}$.
What exactly, if anything, can we say about $\Gamma$? Clearly (in asymptotics) $$\Gamma(n)\ge (1/\alpha+1/\beta)\cdot n$$ But I suspect it is much larger.
$m\alpha+n\beta=k$ is a line in $(m,n)$ space with axis intercepts $\frac k\alpha$ and $\frac k\beta$; together with the positive axes it forms a triangle of area $\frac{k^2}{2\alpha\beta}$, so asymptotically that's the number of admissible $(m,n)$ pairs we should expect in $[0,k]$.