Aug 2026· 5 citations· ⚡ 1 influential· 18 references
Mathematics
Abstract
Let $X\subset\mathbb{R}$ be finite and let $\gamma(t)=(t,f(t))$, where $f$ is strictly convex. We show that \[ J_3(\gamma(X)) =\#\{(x_1,\ldots,x_6)\in X^6:\sum_{i=1}^3\gamma(x_i)=\sum_{i=4}^6\gamma(x_i)\} \ll_{\epsilon}|X|^{3+\epsilon}. \] When specialized to the parabola, our result implies near-optimal estimates for the number of solutions to the diameter-free quadratic Vinogradov system. As a second application, we settle a conjecture from Krishnapur-Kurlberg-Wigman and Bombieri-Bourgain concerning lattice points on dilates of the unit circle. As a third application, we prove that $|A-A|\gg_\epsilon|A|^{5/3-\epsilon}$ and $|A+A|\gg_\epsilon|A|^{8/5-\epsilon}$ for any finite convex sequence $A\subset \mathbb{R}$.
Let $F:\mathbb{R}^3\to\mathbb{R}$ be a polynomial that is irreducible over $\mathbb{R}$ with $\text{deg} F\geq2$. We prove that, for any finite $X\subset Z(F)$ that does not concentrate on affine lines, \[ E(X)=\#\{(a,b,c,d)\in X^4: a+b=c+d\}\ll_{\text{deg} F,\,\epsilon}(\# X)^{2+\epsilon}. \] In particular, this answe...
We fix an integer $d\ge1$ and a symmetric positive-definite matrix $Q\in\mathbb{R}^{d\times d}$. Let $V:\mathbb{R}^d\to\mathbb{R}$ be finite, set \[ Z_\mu:=\int_{\mathbb{R}^d}e^{-V(x)}\,dx\in(0,\infty), \qquad d\mu(x):=Z_\mu^{-1}e^{-V(x)}\,dx, \] and assume that $\mu$ has finite second moment and that \[ x\longmapsto \...
Let $\Gamma\subset\mathbb R^2$ be a compact convex graph and define \[ T(\Gamma) = \inf \left\{ t: \dim_{\mathrm H}(E)>t \Longrightarrow |E+\Gamma|>0 \text{ for every compact }E\subset\mathbb R^2 \right\}. \] For a graph over an interval of positive length the smallest possible value is $T(\Gamma)=1$. We ask whether th...
A. Iosevich, Zhang Li, E. Palsson et al.· 0 citations
Let $X=(X_1,\ldots,X_n)$ have independent coordinates with mean zero, variance one, and $\|X_i\|_{\psi_2}\le K$, and let $H_d=(\mathbb R^n)^{\otimes_2 d}$. Let $L>0$ and let $f:H_d\to\mathbb R$ be convex and $L$-Lipschitz. We prove that, for $0\le t\le c_KLn^{d/2}$, \[ \textsf{P}\left\{ \left\lvert f(X^{\otimes d})-\te...
Let $E\subset\mathbb R^2$ be analytic with $t=\dim_{\mathrm H}E>1$. We prove $\dim_{\mathrm H}\Delta E\ge 4/5+\eta_0$ for an absolute, non-explicit constant $\eta_0>0$, and \[ \dim_{\mathrm H}\Delta E\ge \frac{t^2-2t+5}{2t^2-7t+10} \quad\text{for }1<t\le\frac{11-\sqrt{41}}{4}, \] which reaches $7/8$ at the right endpoi...
Let $X\subset \mathbb C^n$ be a closed pure $d$-dimensional complex analytic set. We associate to $X$ its set of projective asymptotic directions $$ \Sigma_\infty(X) :=\{\ell \in \mathbb P^{n-1}(\mathbb C):\ell\cap C_\infty(X)\ne\{0\}\}, $$ where $ C_\infty(X)$ is the total tangent cone at infinity. We prove the metric...
Nhan D. V. Nguyen· 1 citation· ⚡1
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