Large Sieve Estimates on Arcs of a Circle, Original Research Article

Let 0less-than-or-equals, slantα<βless-than-or-equals, slant2π and let Δimage{eiθ: θset membership, variant[α, β]}. We show that for generalized (non–negative) polynomials P of degree r and p>0, we haveimageless-than-or-equals, slantcτ(pr+1) ∫βα P(eiθ)p dθ, where a1, a2, …, amset membership, variantΔ, 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.