Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2205.07431

Radial projection theorems in finite spaces

Ben Lund·Thang Pham·Vu Thi Huong Thu
View original source ↗Read source document ↗Export metadata ↓

Abstract

Motivated by recent results on radial projections and applications to the celebrated Falconer distance problem, we study radial projections in the setting of finite fields. More precisely, we extend results due to Mattila and Orponen (2016), Orponen (2018), and Liu (2020) to finite spaces. In some cases, our results are stronger than the corresponding results in the continuous setting. In particular, we solve the finite field analog of a conjecture due to Liu and Orponen on the exceptional set of radial projections of a set of dimension between $d-2$ and $d-1$.

Explore related subjects

math.CAmath.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Ben Lund, Thang Pham, Vu Thi Huong Thu. 2022-08-01. Radial projection theorems in finite spaces. https://arxiv.org/abs/2205.07431

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Entropy Expansion for General Polynomial Images of Frostman Random Variables

We prove dyadic entropy expansion for the observables $X+Y$ and $f(X,Y)$ under Frostman nonconcentration hypotheses on a prescribed, possibly dependent law. For an integer $n\ge1$, write $H_n(Z)=H(\lfloor 2^nZ\rfloor)$ for the base-two Shannon entropy at resolution $2^{-n}$; thus $n$ indexes the fineness of the dyadic discretization. For every $0 4/3$, we obtain $\max\{H_n(X+Y),H_n(f(X,Y))\}\ge (\frac{s_1+s_2}{2}+\varepsilon)n-O(1)$ for an explicit $\varepsilon>0$. Here the baseline averages the two Frostman exponents. We also classify the exceptional coordinate representations and construct obstructions for algebraic affine directions with Frostman constants uniform in the scale.

math.CA↗
arXiv2026

On entire solutions of two kinds of quadratic trinomial Fermat type functional equations

The existence of entire solutions to Fermat type differential-difference equations and \(q\)-difference differential equations involving second-order derivatives is investigated by using Nevanlinna theory. The exact forms of the entire solutions to these equations mentioned above are identified. These results represent a generalization and improvement of previous findings obtained by Gong et al. [Bull. Iran. Math. Soc. 51, 17 (2025)]. Furthermore, some examples are provided to illustrate these results.

math.CA↗
arXiv2026

On the prime field spherical restriction conjecture in four dimensions: breaking the Stein-Tomas exponent and applications

Let $p$ be an odd prime. We prove the extension estimate $R_{S_j}^*(2\to r)\lesssim_r 1$ for every nonzero-radius sphere $S_j\subseteq\mathbb{F}_p^4$ and every $r\geq 16/5$, uniformly in $p$ and $j$. This improves the Stein--Tomas exponent $10/3$ established by Iosevich and Koh (2008). To prove this result, we develop a new method that combines horizontal slicing, cancellation in an affine fourth-moment expansion, a rich-plane decomposition, and incidence estimates. We also formulate a localized spherical restriction/extension conjecture that predicts the sharp dependence of the restriction norm on the size of the physical support. This conjecture implies the spherical extension estimates $R_{S_j}^*(2\to r)\lesssim_r 1$ for every $r>3$, and yields almost-every-pin distance estimates at the conjectured Erdős--Falconer exponent in four dimensions, up to an arbitrarily small power loss in the set-size hypothesis. Using the same method, we improve the bounds supplied by Fourier decay and Plancherel at intermediate support scales and derive new almost-every-pin distance estimates in $\mathbb{F}_p^4$.

math.CA↗