  We prove an $\ell^2L^p$ decoupling inequality for functions on $\mathbb{R}$
with Fourier transform supported in a neighborhood of short Dirichlet sequences
$\{\log n\}_{n=N+1}^{N+N^{1/2}}$, as well as sequences with similar convexity
properties, in the sharp range $2\leq p\leq 6$. The proof exploits the wave
packet structure of functions with frequency support near an arithmetic
progression.
