Higher uniformity of arithmetic functions in short intervals I. All intervals

Abstract We study higher uniformity properties of the Möbius function, the von Mangoldt function, and the divisor functions dₖ on short intervals (X, X+H] with X^ + H X^1- for a fixed constant 0 0. More precisely, letting ^ and dₖ^ be suitable approximants of and dₖ and ^ = 0, we show for instance that, for any nilsequence F (g (n) ), we have align*ₗ < ₍ ₗ+₇ (f (n) -f^ (n) ) F (g (n) ) H ^-A X align* when = 5/8 and f \, , dₖ\ or = 1/3 and f = d₂. As a consequence, we show that the short interval Gowers norms \|f-f^ \|ₔ⌁ (ₗ, ₗ+₇] are also asymptotically small for any fixed s for these choices of f,. 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 L². 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 I₂ sums.