The number of integer points with a given norm in the integer grid $\mathbb{Z} \times \mathbb{Z}$ may be calculated via the generating function
$$\theta\_3(q)^2= \left(\sum\_{n \in \mathbb{Z}} q^{n^2}\right)^2 = 1+4 q + 4 q^2 + 4 q^4 + 8 q^5 + \cdots,$$ 
which is a so called (squared) Jacobi function. Similarly the generating function for the shifted $(\mathbb{Z}+a)\times(\mathbb{Z}+b)$ is given by (say $\Theta(a,b)$)
$$\Theta(a,b)=\left(\sum\_{n \in \mathbb{Z}} q^{(n+a)^2}\right)\left(\sum\_{n \in \mathbb{Z}} q^{(n+b)^2}\right)$$ 
which can also be written as a function as Jacobi function of two variables.


For example, if $(a,b)=(\sqrt{2}/3,3/10)$, we have


$$\Theta(a,b) \approx q^{1.91222}+q^{0.769413}+q^{0.712222}+q^{0.369413}+q^{0.312222}$$


My question is if the following problem (or anything related) has been studied: what is the value of (a,b) that makes
(i) all coefficients in the expansion of $\Theta(a,b)$ equal to one and (ii) the **difference between successive exponents** in $\Theta(a,b)$ the largest possible.



"Sparse" Theta Series
Won't the following argument show that the difference between successive exponents can never be bounded away from zero no matter how clever you try to be in selecting $(a,b)$?


The idea is to consider pairs of integers $(n,m)$ such that $na-mb$ is close to zero.


1. If the ratio $a/b$ is rational then we can find integers $n,m$ such that $na=mb$. But in that case
$$(n+a)^2+(-m+b)^2=n^2+m^2+a^2+b^2+2(na-mb)=n^2+m^2+a^2+b^2$$
and also
$$
(-n+a)^2+(m+b)^2=\cdots=n^2+m^2+a^2+b^2
$$
meaning that the theta series has coefficients $>1$.
2. On the other hand if the ratio $a/b$ is irrational then, to a given $\epsilon>0$, we can find integers $n,m$ such that
$$
|na-mb|<\epsilon.
$$
This is because the additive group generated by $a$ and $b$ is then a dense subset of $\Bbb{R}$. But, reusing the above points, we see that
$$
\begin{aligned}
||(n+a,-m+b)||^2-||(-n+a,m+b)||^2&=(n^2+m^2+a^2+b^2)+2(na-mb)\\
&-(n^2+m^2+a^2+b^2)+2(na-mb)\\
&=4(na-mb),
\end{aligned}
$$
which is $<4\epsilon$.


A geometrical motivation for finding these points came from the observation that when $\pm(n,-m)$ is nearly orthogonal to $(a,b)$, we are bound to get two points yielding nearly equal exponents of $q$.


