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 answers a question of Bourgain and Demeter concerning finite subsets of the unit sphere.
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...
Adam Cushman, C. Demeter, Shu-Kun Wu· 5 citations· ⚡1
In this note, we prove uniform upper bounds for the volume of the level set $$\{x\in\Omega: c\le f(x)0,$$ for strictly convex and hyperbolic functions $f$ defined on a convex domain $\Omega\subset\mathbb{R}^n$ ($n\ge 2$). In particular, under a Hessian lower bound $D^2f\ge I_n$, we obtain the sharp volume bound $$|S^{n...
Let $3\leq n\leq7$, $2\leq k\leq n-1$, and $m=n-k-1$. We prove that a complete, connected, noncompact spin manifold $(M^n,g)$ with scalar curvature $R_M\geq k(k-1)$ is isometric to $\mathbb{S}^k\times\mathbb{T}^m_\Lambda\times\mathbb{R}$ if it admits a smooth proper map of nonzero degree to $\mathbb{S}^k\times\mathbb{T...
Consider the random lacunary series $f(z) = \sum_{k=1}^\infty \frac{\xi_k}{\sqrt{k}} \, z^{2^k}$ on the unit disk, where $\{ \xi_k \}$ are independent standard complex Gaussian random variables. We show that almost surely, a.e. $\zeta \in \partial \mathbb{D}$ is a Plessner point of $f$, yet the image of every Stolz ang...
The Bernstein problem for the affine maximal type equation \[ \sum_{i,j=1}^n f^{ij} w_{ij}=0,\qquad w\equiv \left[\det\left(\frac{\partial^2 f}{\partial x_i\partial x_j}\right)\right]^a,\quad x\in\Omega\subset\mathbb R^n, \] is a central problem in affine geometry. It originates from Chern's conjecture on entire locall...
We prove that the bound \[ \max \{ |16A|,|16f(A)| \} \gg_m |A|^{\frac{3}{2}+\frac{1}{162}} \] holds for any polynomial $f$ with degree $m \geq 2$ and any finite $A \subset \mathbb R$. This shows that the classical Jarn\'{i}k obstruction to growth beyond exponent $3/2$, which occurs for general strictly convex functions...
Michalis Kokkinos, O. Roche-Newton· 0 citations
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