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

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 and sum-product type phenomena, and we extend the method to $(u,s)$-Salem sets for even moments $u\ge4$.