We study higher uniformity properties of the Möbius function
$\mu $, the von Mangoldt function
$\Lambda $, and the divisor functions
$d_k$ on short intervals
$(X,X+H]$ with
$X^{\theta +\varepsilon } \leq H \leq X^{1-\varepsilon }$ for a fixed constant
$0 \leq \theta < 1$ and any
$\varepsilon>0$.
More precisely, letting
$\Lambda ^\sharp $ and
$d_k^\sharp $ be suitable approximants of
$\Lambda $ and
$d_k$ and
$\mu ^\sharp = 0$, we show for instance that, for any nilsequence
$F(g(n)\Gamma )$, we have
$$\begin{align*}\sum_{X < n \leq X+H} (f(n)-f^\sharp(n)) F(g(n) \Gamma) \ll H \log^{-A} X \end{align*}$$
when
$\theta = 5/8$ and
$f \in \{\Lambda , \mu , d_k\}$ or
$\theta = 1/3$ and
$f = d_2$.
As a consequence, we show that the short interval Gowers norms
$\|f-f^\sharp \|_{U^s(X,X+H]}$ are also asymptotically small for any fixed s for these choices of
$f,\theta $. As applications, we prove an asymptotic formula for the number of solutions to linear equations in primes in short intervals and show that multiple ergodic averages along primes in short intervals converge in
$L^2$.
Our innovations include the use of multiparameter nilsequence equidistribution theorems to control type
$II$ sums and an elementary decomposition of the neighborhood of a hyperbola into arithmetic progressions to control type
$I_2$ sums.