A sharp point-sphere incidence bound for $(u, s)$-Salem sets

Abstract: We establish a sharp point-sphere incidence bound in finite fields for point sets exhibiting controlled additive structure. Working in the framework of ((4,s))-Salem sets, which quantify pseudorandomness via fourth-order additive energy, we prove that if (P\subset \mathbb{F}_q^d) is a ((4,s))-Salem set with (s\in \big( \frac{1}{4}, \frac{1}{2} \big]) and (|P|\ll q^{ \frac{d}{4s}}), then for any finite family (S) of spheres in (\mathbb{F}_q^d), [ \bigg| I(P,S)-\frac{|P||S| }{q} \bigg| \ll q^{\frac{d}{4}}\,|P|^{1-s}\,|S|^{\frac{3}{4}}. ] This estimate improves the classical point-sphere incidence bounds for arbitrary point sets across a broad parameter range. The proof combines additive energy estimates with a lifting argument that converts point-sphere incidences into point-hyperplane incidences in one higher dimension while preserving the ((4,s))-Salem property. As applications, we derive refined bounds for unit distances, dot-product configurations, and sum-product type phenomena, and we extend the method to ((u,s))-Salem sets for even moments (u\ge4).