arXiv subjects

Adam Cushman

Publications and source records attributed to Adam Cushman.

3 recordsLinked to original sources

arXiv2026

Near optimal three-fold additive energy bound for points on convex curves

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 $A\subset \mathbb{R}$.

math.CA↗

arXiv2026

A note on the sum-product problem for fractal sets

Utilising recent advances in incidence geometry for balls and tubes, and advances in sum-product theory in the discrete setting, we show that for $0 < s \leq 1/2$ and for any $A \subset \mathbb{R}$ with Hausdorff dimension $s$, either the upper-box dimension of $AA$, or the lower-box dimension of $A+A$ must be at least $29s/23$. We obtain the slightly better bound of $33 s / 26$ when we replace the sum-set with the smoother difference-set.

math.CA↗

arXiv2025

A Note on the Sum-Product Problem and the Convex Sumset Problem

We provide a new exponent for the Sum-Product conjecture on $\mathbb{R} $. Namely for $A \subset \mathbb{R}$ finite, \[ \max \left\{ \left\lvert A+A \right\rvert , \left\lvert AA \right\rvert \right\} \gg_{\epsilon} \left\lvert A \right\rvert ^{\frac{4}{3} + \frac{10}{4407} - \epsilon} .\] We also provide new exponents for $A \subset \mathbb{R} $ finite and convex, namely \[ \left\lvert A+A \right\rvert \gg_{\epsilon} \left\lvert A \right\rvert ^{\frac{46}{29} - \epsilon}, \] and \[ \left\lvert A-A \right\rvert \gg_{\epsilon} \left\lvert A \right\rvert ^{\frac{8}{5} + \frac{1}{3440} -\epsilon} .\]

math.CO↗