Restricted projection exceptional-set conjecture in R3

Prove that for every Borel set A subset of R^3 with Hausdorff dimension t in [0,3] and every s in [0,min{t,1}], the exceptional set of angles theta in [0,2pi) for which the orthogonal projection rho_theta(A) has Hausdorff dimension less than s satisfies dim{theta: dim rho_theta(A)<s} <= max{3s/2 - t/2,0}.

Background

The paper relates sine-wave Furstenberg sets to restricted orthogonal projections from R3 onto the one-dimensional subspaces spanned by (cos theta, sin theta, 1). This duality motivates an analogue, in the restricted-projection setting, of Oberlin’s conjecture for planar projections.

Conjecture 1.4 proposes an upper bound for the Hausdorff dimension of the exceptional set of projection directions. The paper notes that several boundary cases are known, including the cases s=t when t<=1, s=1 when t>=1, and s=t/3, but the full interpolation remains conjectural.

References

Following this approach, we formulate an analogue of Oberlin’s conjecture in the restricted projection setting of R3. Conjecture 1.4. Let t ∈ [0, 3] and let s ∈ [0, min {t, 1}]. If A ⊆ R3 is a Borel set with dim A = t, then dim {θ ∈ [0, 2π) : dim ρθ (A) < s} ≤ max{ 3s2 − t2 , 0}.

— Incidence bounds related to circular Furstenberg sets  (2502.10686 - Green et al., 15 Feb 2025) in Conjecture 1.4, Section 1.2, page 5