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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 4 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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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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