For ease of notation, below when I write $|x| \mod N$, I am referring to the absolute value of the integer with smallest absolute value in the same residue class as $x$. For example $|6| \mod 7$ is $1$.
A natural corollary of the simultaneous version of the Dirichlet's approximation theorem is the following statement. $$\forall a_1,a_2,\cdots,a_d \in \mathbb{Z}_N, \exists p \in \mathbb{Z}_N^* : |pa_i| \leq N^{1 - \frac{1}{d}} \mod N.$$ For $d = 2$, the bound $|pa_i| \leq \lfloor\sqrt{N}\rfloor$ is tight for every $N$. That is to say, for every $N$: $$\exists a_1, a_2 : \min_{p}\max(|a_1p|,|a_2p|) = \lfloor\sqrt{N}\rfloor \mod N.$$ For $d = 3$ the bound of $\lfloor N^{2/3} \rfloor$ is only tight for particular values of $N$. I am particularly interested in two questions here.
For which values of $N$ is this bound tight? Computationally this set of $N$s begins: $$2, 4, 5, 7, 8, 10, 11, 14, 18, 26, 27, 30, 31, 63, 64, 68, 69, 70, 76,\ldots.$$ As an initial observation, the bound appears to be tight when $N$ is a cube or when $N + 1$ is a cube. In this set, there are also $N$, $N+1$ pairs which don't fit this criteria, but I am not sure how to characterize those.
Can the $\lfloor N^{2/3} \rfloor$ bound be tightened per $N$? I would be interested in any result that can tighten the bound for particular classes of $N$s or even a non-trivial asymptotic lower bound on: $$B_N = \max_{(a_1,a_2,a_3)}\min_{p}\max_{i}|a_i p| \mod N.$$ Starting with $N = 2$, the sequence $\lfloor N^{2/3} \rfloor - B_N$ begins: $$0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 2, 2,\ldots$$ With record values occurring at $2, 3, 23, 118, 217, 611, \ldots$
None of these sequences appear in the OEIS, I have not found any results in the literature, and I am stuck on trying to refine classic proofs of the simultaneous version of the Dirichlet's approximation theorem for this specific problem. Any results, observations, or pointers to relevant literature would be appreciated.