By a classical result of Singer (1938), for a prime number $p$ the cyclic group $C_n$ of order $n=1+p+p^2$ contains a subset $D$ of cardinality $|D|=1+p$ such that $DD^{-1}=C_n$. Such set $D$ is called a *difference set*. The cardinality restrictions imply that each non-unit element $x\in C_n$ can be uniquely written as the difference $x=a-b$ for $a,b\in D$.

Identify $C_n$ with a subgroup of the unit circle $\mathbb T=\{z\in C_n:|z|=1\}$ on the complex plane. Let $D$ be a subset of $C_n\subset \mathbb T$. A 2-element set $\{a,b\}\subset D$ is called a *gap* in $D$ if there exists an arc $A\subset \mathbb T$ with end-points $a,b$ such that $A\cap D=\{a,b\}$. The real number $|a-b|$ is called the *diameter* of the gap $\{a,b\}$.

**Problem 1.** What is the largest possible diameter of a gap in a difference set in a cyclic group $C_n$?

More precisely:

**Problem 2.** Is there $\epsilon>0$ such that for every prime number $p$ the cyclic group $C_n\subset\mathbb T$ of order $n=1+p+p^2$ contains a difference set $D$ with a gap of diameter $>\varepsilon$?