Sparse exponential sums with low sidelobes

Abstract

Let k₁, …, k_N be integers satisfying 0 ≤ k_j < N_p for positive interges N and p. For infinitely many integers N a construction is given for sets of integers {k₁,…, k_N} such that

$$\mathop{{\sup }}\limits_{{ - 1/2 < x \leqslant 1/2}} \left( {|\sum\limits_{{j = 1}}^{N} {{{e}^{{2\pi i{{k}_{j}}x}}}} | - |\frac{{\sin \pi Npx}}{{p{\mkern 1mu} \sin \pi x}}|} \right) = O\left( {{{N}^{{1/2}}}} \right) $$

. The N are of the from ${{p}^{{M + g(M)}}}$ where $(M){\mkern 1mu} = {\mkern 1mu} O{\mkern 1mu} ({{p}^{{M/2}}})$.