Large Sieve Estimates on Arcs of a Circle

Abstract Let 0⩽α = def {eiθ: θ∈[α, β]}. We show that for generalized (non–negative) polynomials P of degree r and p>0, we have ∑ j =1 m |P(a j )| p | a j − e iα | | a j − e iβ |+ β−α pr +1 2 1/2 ⩽cτ(pr+1) ∫ β α |P(e iθ )| p dθ, where a1, a2, …, am∈Δ, c is an absolute constant (and, thus, it is independent of α, β, p, m, r, P, {aj}) and τ is an explicitly determined constant which measures the number of points {aj} in a small interval. This implies large sieve inequalities for generalized (non–negative) trigonometric polynomials of degree r on subintervals of [0, 2π]. The essential feature is the uniformity of the estimate in α and β.