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

We study higher uniformity properties of the Möbius function μ , the von Mangoldt function Λ , and the divisor functions dₖ on short intervals $(X,X+H]$ with Xθ +ε ≤ H ≤ X1-ε for a fixed constant 0 ≤ θ < 1 and any ε>0 . More precisely, letting Λ ^ and dₖ^ be suitable approximants of Λ and dₖ and μ ^ = 0 , we show for instance that, for any nilsequence F(g(n)Γ ) , we have align*∑X < n ≤ X+H (f(n)-f⁽n)) F(g(n) Γ) H log-A X align* when θ = 5/8 and f ∈ \Λ , μ , dₖ\ or θ = 1/3 and f = d₂ . As a consequence, we show that the short interval Gowers norms \|f-f^ \|Uˢ(X,X+H] are also asymptotically small for any fixed s for these choices of f,θ . 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.