Remove the lattice term from the Furstenberg lower bound

Determine whether the term \(\frac{3s+t}{2}\) can be removed from the planar Furstenberg lower bound when additional conditions are imposed on the intersections \(E\cap\ell\), thereby improving the estimate beyond the sharp general bound in the presence of such conditions.

Background

The classical planar Furstenberg theorem gives the lower bound min⁡{s+t,(3s+t)/2,s+1}\min\{s+t,(3s+t)/2,s+1\} for a set EE intersecting a family of lines in sets of dimension at least ss. The middle term is realized by a lattice construction. The paper explicitly raises the unresolved question of whether additional structural assumptions on the line intersections can eliminate this term.

References

It is a natural problem to ask whether, under some additional conditions on E\cap\ell, this term can be removed.

— Spread-out Furstenberg sets I: the sticky case  (2610.13677 - Orponen et al., 9 Oct 2026) in Section 1, subsection “Optimal Furstenberg estimates”