Let $\langle x\rangle$ denote the fractional part of a real number $x$ (i.e. $\langle x \rangle := x- \lfloor x\rfloor $, where $\lfloor x\rfloor $ is the greatest integer less than or equal to $x$).

Let $\alpha \in \mathbb R$ be irrational and let $S:=\{n\in \mathbb Z: \langle n\alpha \rangle \in (0,1/4)\}$. The upper (and lower) density of $S$ is $1/4$; this is a consequence of Weyl's theorem on uniform distribution. Also, $S-S\subseteq \{n\in \mathbb Z: \langle n\alpha\rangle \in (3/4,1)\cup [0,1/4)\}$.

To see that $S-S$ does not contain an infinite arithmetic progression $\{a+bn:n\in \mathbb N\}$, note that $b\alpha$ is irrational if $b\in \mathbb Z\setminus \{0\}$, so the values $\langle (a+bn)\alpha \rangle$ are dense in $[0,1]$. So if $S-S$ contained an infinite AP, the values $\{\langle n\alpha \rangle:n\in S-S\}$ would be dense in $[0,1]$, but $\langle n\alpha\rangle \in (3/4,1)\cup [0,1/4)$ for $n\in S-S$.

This example $S$ is a Bohr neighborhood in $\mathbb Z$. Generally, if you want an example or counterexample of some structure in $S-S$, where $S$ has positive upper density, it's natural to look among Bohr neighborhoods: Følner ZBL0058.02302) proved that if $S$ has positive upper Banach density, then $S-S$ contains (up to upper Banach density 0) a Bohr neighborhood of $0$. Since every Bohr neighborhood $B$ of $0$ contains a set of the form $B'-B'$, where $B'$ is a Bohr neighborhood, $S-S$ itself is not too far from containing a difference set of a Bohr neighborhood.

Ruzsa's section Sumsets and structure in ZBL1221.11026 and Hegyvári and Ruzsa's article ZBL1333.05042 are both good references on the relationship between Bohr sets and difference sets.

Let $\langle x\rangle$ denote the fractional part of a real number $x$ (i.e. $\langle x \rangle := x- \lfloor x\rfloor $, where $\lfloor x\rfloor $ is the greatest integer less than or equal to $x$).

Let $\alpha \in \mathbb R$ be irrational and let $S:=\{n\in \mathbb Z: \langle n\alpha \rangle \in (0,1/4)\}$. The upper (and lower) density of $S$ is $1/4$; this is a consequence of Weyl's theorem on uniform distribution. Also, $S-S\subseteq \{n\in \mathbb Z: \langle n\alpha\rangle \in (3/4,1)\cup [0,1/4)\}$.

To see that $S-S$ does not contain an infinite arithmetic progression $\{a+bn:n\in \mathbb N\}$, note that $b\alpha$ is irrational if $b\in \mathbb Z\setminus \{0\}$, so the values $\langle (a+bn)\alpha \rangle$ are dense in $[0,1]$. So if $S-S$ contained an infinite AP, the values $\{\langle n\alpha \rangle:n\in S-S\}$ would be dense in $[0,1]$, but $\langle n\alpha\rangle \in (3/4,1)\cup [0,1/4)$ for $n\in S-S$.

This example $S$ is a Bohr neighborhood in $\mathbb Z$. Generally, if you want an example or counterexample of some structure in $S-S$, where $S$ has positive upper density, it's natural to look among Bohr neighborhoods: Følner ZBL0058.02302 proved that if $S$ has positive upper Banach density, then $S-S$ contains (up to upper Banach density 0) a Bohr neighborhood of $0$. Since every Bohr neighborhood $B$ of $0$ contains a set of the form $B'-B'$, where $B'$ is a Bohr neighborhood, $S-S$ itself is not too far from containing a difference set of a Bohr neighborhood.

Ruzsa's section Sumsets and structure in ZBL1221.11026 and Hegyvári and Ruzsa's article ZBL1333.05042 are both good references on the relationship between Bohr sets and difference sets.