This question is related to the question I asked earlier.

For a natural number $n$, a set $D$ of integer numbers is called a *$n$-cyclic difference set* if each integer number $x\notin n\mathbb Z$ can be uniquely represented as the difference $x\equiv a-b \mod n$

for a unique pair $(a,b)\in D\times D$. For example, $\{0,1,3,6\}$ is a $13$-cyclic difference set.

For every power $q=p^k$ of a prime number $p$, Singer (1938) constructed a $(1+q+q^2)$-cyclic difference set $D$ of order $|D|=q+1$. A (still unproved) Prime Power Conjecture says that a number $n$ admitting an $n$-cyclic difference set is of the form $n=1+q+q^2$ for some power $q$ of a prime number. The Singer's cyclic difference sets play an important role in Additive Combinatorics.

We are interested in evaluating the largest gap in an $n$-cyclic difference set. A pair $(a,b)$ of numbers is called a *gap* in a set $A\subset \mathbb Z$ if the intersection $[a,b]\cap A$ coinsides with the doubleton $\{a,b\}$. The number $|a-b|$ is called the *length* of the gap. For a subset $A\subset\mathbb Z$ let $gap(A)$ be the largest lenght of a gap in $A$. For example, $g(\{0,1,3,6\})=|6-3|=3$.

**Problem 1.** What is the largest possible gap in an $n$-cyclic difference set $D$?

A more precise question:

**Problem 2.** Is it true that for any positive constant $C$ there exists an $n$-cyclic difference set $D$ with $gap(D)>C\cdot \sqrt{n}$?

**Remark 1.** In fact, Problem 2 was implicitely asked by Leech in 1956 and discussed by Golay in 1972. So, maybe this problem has been already solved, in which case I would greatly appreciate a corresponding reference.

**Remark 2.** Answering my preceding question, @Lucia proved that for every $\varepsilon>0$ there exists a number $n_\varepsilon$ such that $gap(D)<\varepsilon \cdot n$ for any $n$-cyclic difference set $D$ with $n \ge n_\varepsilon$. So, gaps in cyclic difference sets cannot be very large.

I can prove that for any prime power $q=p^k$ each $(1+q+q^2)$-cyclic difference set $D$ has $gap(D)\ge q+\sqrt{q}-1$, but it is very far from $gap(D)>Cq$.