Restricted projections in positive characteristic via Fourier extension and restriction estimates

Le Quang Ham; Do Trong Hoang; Le Quang Hung; Doowon Koh; Thang Pham

Paper

Restricted projections in positive characteristic via Fourier extension and restriction estimates

Abstract

Let

and

be the

-dimensional vector space over a finite field of order

, where

is an odd prime power. Let

be the set of lines through the origin intersecting the slice

, where

and

. For

and

, we study the exceptional sets

with their respective natural ranges of

. Using discrete Fourier analysis together with restriction/extension estimates for cone and sphere-type quadrics over finite fields, we obtain sharp upper bounds (up to constant factors) for

and

, with separate analyses for the cases

. The bounds exhibit arithmetic-geometric dichotomies absent in the full Grassmannian: the quadratic character of

and the parity of

determine the size of the exceptional sets. As an application, when

, there exists a positive proportion of elements

such that the pinned dot-product sets

have cardinality

. We further study analogous families arising from the spheres of radii

and

, and, by combining the results, recover the known estimates for projections over the full Grassmannian, complementing a result of Chen (2018).