Effective form of the sharp Furstenberg set estimate in the plane: explicit parameters, losses, and structural hypotheses in the Ren–Wang discretized inequality

The sharp $(s,t)$-Furstenberg set conjecture in the plane was resolved in 2023 by Ren and Wang (arXiv:2308.08819): every planar set meeting a $t$-dimensional family of lines in sets of Hausdorff dimension at least $s$ has dimension at least $\min\{s+t,(3s+t)/2,s+1\}$. The heart of the proof is a discretized inequality (Theorem 4.1 there): for every $\varepsilon>0$ there exist $\eta=\eta(\varepsilon,s,t)>0$ and $\delta_0=\delta_0(s,t,\varepsilon)>0$ such that every nice configuration of dyadic cubes and tubes at scales $\delta<\delta_0$ satisfies a sharp incidence lower bound. All of these parameters are non-effective: $\eta$ is made independent of $t$ by two compactness (finite-subcover) arguments; scale thresholds are chosen by pure existence; the key incidence lemma carries unspecified $\delta^{-5\varepsilon}$ and $|\log\Delta|$ losses under a uniform-branching hypothesis; and the proof black-boxes a projection input of Orponen–Shmerkin whose constants are non-explicit — the same authors describe the $\varepsilon$-losses in this chain of results as 'tiny and non-explicit'. This problem asks for the effective version: prove the discretized inequality with explicitly computable $\eta(\varepsilon,s,t)$, $\delta_0(s,t,\varepsilon)$, and constants via constructive arguments (effectivizing or bypassing the Orponen–Shmerkin input); give effective forms of the intermediate multiscale incidence lemmas with tracked losses and thresholds; and quantify how far the uniform-branching and related structural hypotheses can be weakened. The effective version of the estimate is open.

Background

A Furstenberg set in the plane is a set that is large on many lines. Writing $\mathbb{A}(2,1)$ for the space of all affine lines in $\mathbb{R}^2$ (a two-dimensional manifold, parametrized for instance by slope and intercept) and $\dim_H$ for Hausdorff dimension (the standard fractal dimension, defined via covers by sets of small diameter), a set $E\subset\mathbb{R}^2$ is an $(s,t)$-Furstenberg set, for $s\in(0,1]$ and $t\in(0,2]$, if there is a family of lines $\mathcal{L}\subset\mathbb{A}(2,1)$ with $\dim_H\mathcal{L}\ge t$ such that $\dim_H(E\cap\ell)\ge s$ for every $\ell\in\mathcal{L}$. The problem of lower-bounding $\dim_H E$ originated in H. Furstenberg's unpublished work on $\times2,\times3$-invariant sets (the case $t=1$, recorded by Wolff in the lecture notes cited below) and was posed for general $(s,t)$ in the late 1990s. The conjectured sharp bound, proved by Ren and Wang in the source paper of this candidate, is $\dim_H E\ \ge\ \min\Big\{s+t,\ \frac{3s+t}{2},\ s+1\Big\},$ and all three expressions are sharp, witnessed respectively by a product of two Cantor sets, by a lattice example discovered by Wolff (sharp for the classical Szemerédi–Trotter point–line incidence bound), and by a product of an interval with a Cantor set. Before 2023, the extreme regimes ($t\le s$ and $s+t\ge 2$) and a few $\varepsilon$-improvements were known, but the middle range $s<t<2-s$ resisted all attacks. The same paper also deduces from the discretized inequality below a sharp exceptional-set estimate for orthogonal projections of planar sets (Theorem 1.2 there).

Like most problems in fractal geometry, the question is attacked through a finitary discretization, and the discretized form is what applications actually use. Fix dyadic scales $\delta=2^{-n}\le\Delta\le 1$. Write $\mathcal{D}_\delta$ for the half-open dyadic $\delta$-cubes in $[0,1)^2$ and $\mathcal{T}_\delta$ for the dyadic $\delta$-tubes (essentially $\delta$-neighborhoods of lines, with a dyadic grid on the slope–intercept parametrization). For a bounded set $A$, $|A|_\delta$ denotes the dyadic $\delta$-covering number, the number of dyadic $\delta$-cubes meeting $A$. A bounded set is a $(\delta,s,C)$-set if it is spread out at every scale $\delta\le r\le 1$: no ball of radius $r$ meets more than a $C\,r^{s}$ fraction of its $\delta$-covering (a discretized Frostman non-concentration condition); the closely related Katz–Tao variant replaces this relative bound by the absolute one $|A\cap B(x,r)|_\delta\le C(r/\delta)^{s}$, a weaker condition when $A$ is much smaller than a full $\delta^{-s}$-set. A pair $(P,\mathcal{T})\subset\mathcal{D}_\delta\times\mathcal{T}_\delta$ is a $(\delta,s,C,M)$-nice configuration if every cube $p\in P$ is associated with a subfamily $\mathcal{T}(p)\subset\mathcal{T}$ of tubes through $p$ that is itself a $(\delta,s,C)$-set of tubes (under the slope parametrization) with $|\mathcal{T}(p)|\sim M$. The discretized Furstenberg set estimate (Theorem 4.1 of the source paper) states: for every $\varepsilon>0$ there exist $\eta=\eta(\varepsilon,s,t)>0$ and $\delta_0=\delta_0(s,t,\varepsilon)>0$ such that for every dyadic $\delta<\delta_0$, every $(\delta,s,\delta^{-\eta},M)$-nice configuration $(P,\mathcal{T})$ with $P$ a $(\delta,t,\delta^{-\eta})$-set obeys $|\mathcal{T}|_\delta\ \gtrsim_\varepsilon\ \delta^{-\min\{t,\,(s+t)/2,\,1\}+\varepsilon}\,M,$ where $\gtrsim_\varepsilon$ suppresses a constant depending only on $\varepsilon,s,t$. A limiting argument recovers the dimension theorem, and the finitary form is what feeds the applications to projections and sum-product.

The proof combines two tracks. For point sets $P$ that are almost AD-regular — AD-regular meaning that the mass of $P$ in any ball of radius $r\in[\delta,1]$ centered at a point of $P$ is comparable to $r^{t}|P|$, and 'almost' allowing a $\delta^{-\varepsilon}$ distortion — the estimate is reduced to a projection input of Orponen–Shmerkin (Corollary 4.9 of their paper on the $ABC$ sum-product problem — the question of whether, for Borel sets $A,B,C\subset\mathbb{R}$ with $\dim_H B\le\dim_H A<1$ and $\dim_H C>\dim_H A-\dim_H B$, some dilated sum $A+cB$ with $c\in C$ has Hausdorff dimension strictly larger than $A$): for every $0\le u<\min\{(s+t)/2,1\}$ there exist $\epsilon,\delta_0>0$ such that every $(\delta,t,\delta^{-\epsilon})$-regular point set admits many large projections in every $(\delta,s,\delta^{-\epsilon})$-set of directions. That input is itself non-explicit and descends through the $ABC$ sum-product theorem and ultimately through Bourgain's discretized sum-product and projection theorems (finitary analogues, for $\delta$-discretized sets, of the facts that a fractal set of dimension greater than $1/2$ grows under pointwise sums or products, and has large orthogonal projections in almost every direction). For semi-well-spaced $P$ — spread out at some coarse scale $\Delta$ and Katz–Tao non-concentrated below it — Ren–Wang prove a sharp incidence bound via the high-low method: an incidence count is Fourier-decomposed into a low-frequency part controlled by the geometry of the coarse scale and a high-frequency part controlled by $L^2$-orthogonality, iterated across scales. The general case is reduced to these two by a multiscale decomposition: the branching function of $P$ (the profile of the log-cardinalities of $P$ across the dyadic scales $\Delta_j=2^{-jT}$, $T\in\mathbb{N}$; Definition 6.3) is partitioned into intervals on which $P$ looks either almost AD-regular or semi-well-spaced (Proposition 6.4, with an auxiliary parameter $\tau(\varepsilon,s,u,t)$), and an induction on scales glues the two tracks.

Every quantitative ingredient of this architecture is non-effective, which is the subject of this problem. (a) The smallness parameter $\eta(\varepsilon,s,t)$ is made independent of $t$ by two compactness arguments — the end of the proof of Theorem 6.1, and Remark 6.8 — each covering the compact interval $[s,2-s]$ of $t$-values by finitely many balls $B(t,\eta_t)$ obtained from uncountably many existential statements and taking the minimum over a finite subcover; the value is not extractable from the proof. (b) Scale thresholds are unspecified (the threshold $\Delta_0(\varepsilon)$ inside the proof of Lemma 5.1 is picked as 'a small constant (to be chosen later)'), and intermediate scales are selected there by definite but untracked extremal and pigeonhole choices ('the largest $k$ such that…'); no values are given. (c) The central incidence lemma for the semi-well-spaced track (Lemma 5.1) carries a $\delta^{-5\varepsilon}$ loss, requires $\delta$ below an unspecified threshold depending on $\varepsilon$, and hypothesizes uniform branching: for each $1\le k\le\varepsilon^{-1}$, every cell of the mesh $\mathcal{D}_{\Delta^{k\varepsilon}}$ meeting $P$ contains about the same number of cubes of $P$; the auxiliary energy bound (Lemma 5.2) carries a $|\log\Delta|$ loss. (d) The projection input of Orponen–Shmerkin has its own existential $\epsilon,\delta_0$, described by its authors as 'tiny and non-explicit'. Explicit constants in Furstenberg-type estimates are known only in special structured situations: Guth–Katz–Zahl obtained explicit constants for the discretized sum-product problem, and Di Benedetto–Zahl derived from them the explicit dimension bound $1+1/4536$ for $(1/2,1)$-Furstenberg sets — both in the pre-2023 regime, strictly weaker than the sharp bound.

Problem Statement

The problem is to make the sharp discretized Furstenberg set estimate in the plane effective. Notation as in the background: $\mathcal{D}_\delta$ and $\mathcal{T}_\delta$ are the dyadic $\delta$-cubes and $\delta$-tubes, $|\cdot|_\delta$ is the dyadic $\delta$-covering number, and $(\delta,s,C,M)$-nice configurations and $(\delta,s,C)$-sets are as defined there. There are three requested results with one common objective — a quantitative, compactness-free account of the Ren–Wang estimate.

(1) Core: the effective sharp estimate. Exhibit explicit, computable, total functions $\eta,\delta_0,C:(0,\infty)\times(0,1]\times(0,2]\to(0,\infty)$ of the variable triple $(\varepsilon,s,t)$ — written $\eta(\varepsilon,s,t)$, $\delta_0(s,t,\varepsilon)$ and $C(\varepsilon,s,t)$, with $\varepsilon\in(0,\infty)$, $s\in(0,1]$ and $t\in(0,2]$ — and prove that for all $s\in(0,1]$, $t\in(0,2]$, $\varepsilon>0$, every dyadic $\delta<\delta_0(s,t,\varepsilon)$ and every $(\delta,s,\delta^{-\eta(\varepsilon,s,t)},M)$-nice configuration $(P,\mathcal{T})$ whose point set $P$ is a $(\delta,t,\delta^{-\eta(\varepsilon,s,t)})$-set, $|\mathcal{T}|_\delta\ \ge\ C(\varepsilon,s,t)\,\delta^{-\min\{t,\,(s+t)/2,\,1\}+\varepsilon}\,M.$ The proof must be effective: no compactness, finite-subcover, or pure-existence argument may be used to produce $\eta$, $\delta_0$, $C$ or any auxiliary constant — every constant in the argument is given by an explicit computable rule — and any black-boxed input, in particular the Orponen–Shmerkin projection input (Corollary 4.9 of arXiv:2301.10199) on which the almost-AD-regular track relies, must either be effectivized within the submission or replaced by an effective alternative with its own explicit constants.

(2) Effective multiscale incidence bounds: losses and thresholds. In the course of (1), or independently: give effective versions of the intermediate incidence bounds of the Ren–Wang proof — explicit thresholds $\delta_0(\varepsilon)$ (and the analogous intermediate-scale constraints) for Lemma 5.1 and its applications, and explicit control of the intermediate-scale selections in its proof — and explicit, tracked forms of the losses, replacing the unspecified $\delta^{-5\varepsilon}$ factor in Lemma 5.1 and the $|\log\Delta|$ factor in Lemma 5.2 by explicitly bounded quantities. Where possible, determine whether these losses are sharp.

(3) Minimal structural hypotheses. Quantify how far the structural hypotheses can be weakened while retaining the same final estimates: in particular, how much of the uniform-branching condition in Lemma 5.1 — for each $1\le k\le\varepsilon^{-1}$, every cell of the mesh $\mathcal{D}_{\Delta^{k\varepsilon}}$ meeting $P$ contains about the same number of cubes of $P$ — and of the semi-well-spaced conditions can be dropped, relaxed, or randomized, with explicit dependence of the conclusions on the relaxed hypotheses.

The three parts share one objective, the effectivization of the single Ren–Wang estimate: part (1) is the effective sharp estimate with fully explicit parameters, and parts (2) and (3) quantify the internal losses and the structural assumptions of that same proof.

References

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