Multiple attributes under shuffled differential privacy
The theorem in the paper you refer to only assumes that $\varepsilon&lt;1$ because it simplifies the analysis &amp; the proof — amplification happens regardless of the value of $\varepsilon$. If you look closer at the proof (page 10), you can find a tighter formula: the result holds for any $\varepsilon$ and $\delta$ such that $\mathbb{P}\left[\frac{N_1}{N_2}\geq e^\varepsilon\right]\le\delta$, where $N_1$ and $N_2$ are independent variables sampled from $\text{Bin}\left(n-1,\frac{\gamma}{k}\right)$.
If, instead of bounding that quantity using Chernoff bound and assumptions on $\varepsilon$, you estimate it numerically, you obtain much smaller values of $\varepsilon$ and $\delta$, which show that amplification happens regardless of the value of $\varepsilon$. In a paper I co-authored, we do exactly that; the problem is different but the math ends up being the same. This is Theorem 3, you can see the comparison between the closed-form formula and the numeric estimation on Figure 2.