Discretized Radial Projections in Rd\mathbb{R}^d

We generalize a Furstenberg-type result of Orponen-Shmerkin to higher dimensions, leading to an $\epsilon$-improvement in Kaufman's projection theorem for hyperplanes and an unconditional discretized radial projection theorem in the spirit of Orponen-Shmerkin-Wang. Our proof relies on a new incidence estimate for $\delta$-tubes and a quasi-product set of $\delta$-balls in $\mathbb{R}^d$.Comment: 58 page

Furstenberg sets estimate in the plane

We fully resolve the Furstenberg set conjecture in $\mathbb{R}^2$, that a $(s, t)$-Furstenberg set has Hausdorff dimension $\ge \min(s+t, \frac{s+3t}{2}, s+1)$. As a result, we obtain an analogue of Elekes' bound for the discretized sum-product problem and resolve an orthogonal projection question of Oberlin.Comment: 23 page

Incidence estimates for α\alpha-dimensional tubes and β\beta-dimensional balls in R2\mathbb{R}^2

We prove essentially sharp incidence estimates for a collection of $\delta$-tubes and $\delta$-balls in the plane, where the $\delta$-tubes satisfy an $\alpha$-dimensional spacing condition and the $\delta$-balls satisfy a $\beta$-dimensional spacing condition. Our approach combines a combinatorial argument for small $\alpha, \beta$ and a Fourier analytic argument for large $\alpha, \beta$.Comment: 18 pages, 8 figure

A note on maximal operators for the Schr\"{o}dinger equation on T1.\mathbb{T}^1.

Motivated by the study of the maximal operator for the Schr\"{o}dinger equation on the one-dimensional torus $\mathbb{T}^1$, it is conjectured that for any complex sequence $\{b_n\}_{n=1}^N$, $$\left\| \sup_{t\in [0,N^2]} \left|\sum_{n=1}^N b_n e \left(x\frac{n}{N} + t\frac{n^2}{N^2} \right) \right| \right\|_{L^4([0,N])} \leq C_\epsilon N^{\epsilon} N^{\frac{1}{2}} \|b_n\|_{\ell^2}$$ In this note, we show that if we replace the sequence $\{\frac{n^2}{N^2}\}_{n=1}^N$ by an arbitrary sequence $\{a_n\}_{n=1}^N$ with only some convex properties, then $$\left\| \sup_{t\in [0,N^2]} \left|\sum_{n=1}^N b_n e \left(x\frac{n}{N} + ta_n \right) \right| \right\|_{L^4([0,N])} \leq C_\epsilon N^\epsilon N^{\frac{7}{12}} \|b_n\|_{\ell^2}.$$ We further show that this bound is sharp up to a $C_\epsilon N^\epsilon$ factor.Comment: 13 page