In an arbitrary lattice in $\mathbb{R}^{2}$, can there be a ring about origin, of arbitrarily large area, not containing any lattice point? By a ring I mean an annulus, of such a radius and width so that the area of the annulus is arbitrarily large.
Edit:
Following Michael Lugo's answer for $\mathbb{Z}^{2}$: Any lattice in $\mathbb{R}^{2}$ is generated by two $\mathbb{R}-$linearly independent vectors $v_{1}=(a_{1},b_{1})^{T}$ and $v_{2}=(a_{2},b_{2})^{T}$, and so we are left to find arbitrary large gaps in the sequence of reals of the form $(na_{1}+ma_{2})^{2}+(nb_{1}+mb_{2})^{2}$ for integers $n,m$.
We can always rotate the axes so that one of the vectors is along the $x$ axis, and so we can take $v_{1}=(a_{1},0)^{T}$ with $b_{1}=0$. The vector $v_{2}$ can w.l.o.g be chosen in the first quadrant, and so $a_{2}\geq 0,b_{2}\geq 0$, with $b_{1}=0, a_{1}> 0$. Thus we are seeking large gaps in the sequence $(na_{1}+ma_{2})^{2}+(mb_{2})^{2}$ .
It intuitively seems clear that whatever $a,b,c$ are, we will have arbitrarily large distances between elements of the sequence when $n,m$ are large, and we increase $n$ or $m$ by 1.
 A: Let $R(a, b)$ be the ring about the origin such that $a < x^2 + y^2 < b$.  Clearly this has area $\pi(b-a)$.  So this question is equivalent to asking if there are arbitrarily long gaps in the sequence of integers which can be expressed as sums of two squares.
That sequence begins $0, 1, 2, 4, 5, 8, 9, 10, 13, 16, \ldots$; see OEIS A001481 for more information and more terms in the sequence. So for example $R(2, 4)$ is an annulus of area 2 containing no lattice points; $R(5, 8)$ is an annulus of area 3; $R(20, 25)$ is an annulus of area 5; $R(74, 80)$ is an annulus of area 6; $R(90, 97)$ is an annulus of area 7; and so on.  Looking at this it's hard to guess whether there are arbitrarily large gaps in the sequence.
But the number of integers less than $n$ which can be expressed as a sum of two squares is asymptotically equal to $Cn/\sqrt{\log n}$ for a constant $C$, called the Landau-Ramanujan constant .  In particular, let $k$ be a positive integer.  For sufficiently large $n$, the number of integers less than $n$ which can be expressed as a sum of two squares is less than $n/k$, and therefore there must be a gap in the sequence A001481 (given above) of length at least $k$.  
