For every n≥ 2 and complexity bound E, there is a constant C_n,E such that the following holds. Let 0<delta,lambda≤ 1, let T be a family of unit-length delta-tubes in the unit ball of R^n whose directions are pairwise delta-separated, and let S⊆R^n be a semialgebraic set of complexity at most E. Then #T∈T:| T cap S|≥ lambda| T|≤ C_n,E,delta^-(n-1)lambda^-n| S|.

Guth–Zahl polynomial Wolff axioms conjecture

The Guth–Zahl conjecture asks for the full lower bound governing how many separated tubes can substantially meet a semialgebraic set, a principle formulated by Guth and Zahl in their 2017 work. Its conjectured estimate removes the small factor that earlier results could not eliminate.

Known results

Guth and Zahl (2017; revised 2019) proved substantial linear and multilinear estimates under the polynomial Wolff axioms, but explicitly did not prove the conjecture.
Katz and Rogers (2018) proved the generalized estimate only with a loss of $\delta^{-\varepsilon}$ .
Later work improved consequences in selected dimensions without establishing the full conjecture.

New unrefereed preprint (date not stated)

Gal Binyamini and Yuval Salant claim the full bound for semialgebraic sets meeting many $\delta$ -separated $\delta$ -tubes, removing the $\delta^{\varepsilon}$ loss, and extend it to definable sets in o-minimal structures. They also claim the corresponding algebraic-variety consequence. The result remains unverified.

Current status (as of September 2026): The full conjecture is claimed solved by Binyamini and Salant, but the unrefereed preprint has not been independently verified; earlier published work established only weakened or conditional forms.

Guth–Zahl polynomial Wolff axioms conjecture

Primary source

Additional references

Progress summary

Solutions 0