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Furstenberg sets estimate in the plane
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abstract
We fully resolve the Furstenberg set conjecture in $\mathbb{R}^2$, that a $(s, t)$-Furstenberg set has Hausdorff dimension $\ge \min(s+t, \frac{3s+t}{2}, s+1)$. As a result, we obtain an analogue of Elekes' bound for the discretized sum-product problem and resolve an orthogonal projection question of Oberlin.
Forward citations
Cited by 7 Pith papers
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Projections of self-affine sets onto lines
Under strong pinching and strong irreducibility of the linear parts, every line projection of a self-affine set attains the expected dimension; in the plane, strong irreducibility alone suffices.
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A new proof technique shows distances and orthogonal projections retain at least half of a planar point's Kolmogorov complexity, improving pinned distance dimension bounds to 3/4 s and generalizing Bourgain's theorem.
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Furstenberg set theorem for transversal families of functions
A sharp dimension bound for Furstenberg sets built from transversal families of graphs, with an application to Fourier decay of fractal measures on convex curves.
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Pinned Dot Product Set Estimates
Pinned dot product sets are large when the underlying set has Hausdorff dimension above n/2 with spread translations, or above (n+1)/2 for any translation, proven by reducing dot products to orthogonal projections.
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Two-ends Furstenberg inequality for transversal families and applications to Fourier decay
A simplified two-ends Furstenberg inequality holds for transversal curve families and yields L6 Fourier decay R^{2-5s/2+ε} for s-Frostman measures on convex curves when s≤2/3.
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Sum-product phenomena for Ahlfors-regular sets
Ahlfors regular sets of dimension s ≤ 1/2 satisfy Nδ(A+A) + Nδ(AA) ≥ δ^{-4s/3+η}, the fractal analogue of Solymosi's 4/3 bound.
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Applications of dimension interpolation to orthogonal projections
This survey shows how the Assouad spectrum, intermediate dimensions, and Fourier spectrum yield sharper projection theorems for fractal sets.
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