Perfect Difference Sets (PDS) of order $m+1$ are a set of residues $\{d_1,d_2,\cdots,d_{m+1}\} \pmod{q}$ such that every non-zero residue modulo $q$ can be uniquely represented by $d_i−d_j \pmod{q}$ where $d_i,d_j$ are members of the PDS. Singer [Singer1938] gave the criteria that it is necessary that $q=m^2+m+1$ and sufficient that $m=p^g$, a prime power in order for a PDS to exist.
Singer also showed that there are $\beta = \frac {\varphi(q)} {3g}$ distinct PDS of order $m+1$ where $\varphi$ is the Euler Totient function.
Also, it is a known property of a PDS that if we translated each element by $\delta$ we would obtain another PDS.
By definition, there are $\gamma = q-1$ ways in which a nonzero residue might be represented as a difference modulo $q$.
We have $\beta \lt \gamma$.
Therefore, given $N$ there must be some pairs $d_x - d_y \equiv N \pmod q$ that do not appear in any PDS.
Question: Can we give a criteria for a pair $d_x - d_y \equiv N \pmod q$ to exist in at least one PDS?
References
[Singer1938]: J. Singer, "A Theorem in Finite Projective Geometry and Some Applications to Number Theory," Transactions of the American Mathematical Society, vol. 43, no. 2, p. pp. 377–85, 1938. URL (accessed Dec 1, 2022): https://www.ams.org/journals/tran/1938-043-03/S0002-9947-1938-1501951-4/S0002-9947-1938-1501951-4.pdf