The problem concerns covering the unit square with translates of a specific figure, which I will refer to as a "cross", using as few translates as possible. The difficulty seems to result from the fact that the figure is very concave.
To begin with, let $T^2$ be the unit square $[-1/2,1/2]^2$ with the opposite sides identified. Also, fix a parameter $\varepsilon > 0$. Let us consider a "cross" $K$ defined as: $$ K = \{(x,y) \in T^2 \ : \ |xy| < \varepsilon \} $$ For reasonably small $\varepsilon$ this looks like the coordinate axes "thickened" somewhat. We are interested in the translates of $K$, with the convention that if a part of the translate $K$ "sticks out" of $T^2$ it gets "wrapped around" on the opposite side; thus $(x,y) \in K + (a,b)$ iff $(x,y) \equiv (x'+a,y'+b) \pmod{1}$ for some $(x',y') \in K$. The task is now to cover $T^2$ using as few crosses as possible. It is fairly obvious that the exact number cannot be found, so I am only asking for the asymptotics of this number for sufficiently small $\varepsilon$.
The obvious attempt is to find a suitable rectangle contained in a cross, and use these rectangles to cover the square. One can easily find rectangles of area $\Theta(\varepsilon)$ (i.e. $C \varepsilon$ for a universal constant), so one can cover the square with $\Theta(1/\varepsilon)$ crosses. The area of the largest convex figure that fits into a cross is $\Theta(\varepsilon)$, so this cannot be significantly improved without new insight.
On the other hand, the area of a cross is $\Theta(\varepsilon \log \frac{1}{\varepsilon})$. Thus, one needs at least $\Theta(1/\varepsilon \log \frac{1}{\varepsilon})$ crosses for a cover.
Unfortunately, the two bounds do not agree. Ideally, I would like to know (asymptotically) what's the least cardinality of a cover. More realistically, I would appreciate any argument showing that either of the bounds can be improved.
Motivation: The problem came up when I was considering recurrence rates of generalised polynomials. It is related to asking for an upper bound on the least positive integer $n$ so that $\left< n \alpha \right> \left<n \beta \right> \in (-\varepsilon,\varepsilon)$, where the brackets indicate the fractional part.