We determine the hundredths digit of the real Grothendieck constant by proving $1.773 \leq K_G \leq 1.7799$. Furthermore, numerical heuristics suggest the estimate $K_G \approx 1.779$. We start by simplifying the Gaussian duality theory of $K_G$, based on Hermite projection games and Krivine rounding schemes. The bounds are obtained by a heuristic primal/dual search for these objects, followed by rigorous {\em certification} of their values. The search and certification are performed using AI tools, and in particular the rigorous certificates are very large (analytical reductions to thousands of numerical inequalities).
We give a computer-assisted proof that the real Grothendieck constant is strictly smaller than $1.77987164$. The proof provides a complete interval-arithmetic realization of Ren's five-variable construction. A finite Fourier calculation encloses every scalar coefficient through degree $403$; a fourth-derivative estimat...
We give a computer-assisted proof that the universal real Grothendieck constant satisfies $K_G^{\mathbb R}\le 1.7813$. The construction combines an explicit odd Hermite threshold of degree $11$ with a signed correlation polynomial of degree $51$. A sufficient inverse-majorant inequality is certified by enclosing the sc...
We show that $1.4<K_G^{\mathbb{C}}<1.404898554746$, where $K_G^{\mathbb{C}}$ is the complex Grothendieck constant. The upper bound improves on Haagerup's bound of $1.40490913\ldots$ from 1987, and the lower bound improves on Davie's bound of $1.33807$ from 1984. The upper bound combines ideas from the real and noncommu...
Steven Heilman, Chris Jones, Giulio Malavolta· arXiv.org· 0 citations
We prove the Falconer-type bound $\dim_{\mathrm H}F\geq u+1$ for planar circular $(u,v)$-Furstenberg sets whenever $0<u\leq1$, $0<v\leq3$, and $2u+v\geq3$. The same bound holds for smooth cinematic curve families. This resolves the circular Furstenberg conjecture of Green, Harris, Ou, Ren, and Tammen in its full Falcon...
We determine the exact strong-type Fourier restriction range for the unit sphere $\mathbb{S}^{d-1}$ on the class of $O(d-k)\times O(k)$-invariant functions, for every $d\geq4$ and $2\leq k\leq d-2$. This includes the critical estimates beyond the biradial Stein--Tomas endpoint left open by Mandel and Oliveira e Silva....
Shinya Kinoshita, Juyoung Lee, D. O. E. Silva et al.· 0 citations
We prove that the Tur\'an density of the tetrahedron $K_4^{(3)}$ satisfies $\pi(K_4^{(3)}) \le 14993367693127837/26880000000000000<0.557789$, improving Baber's upper bound of $0.5615$ and closing about $62\%$ of the gap to the conjectured value $5/9$. The proof uses an exact seven-vertex flag-algebra certificate incorp...
Gyeong-Won Jeong, Seonghun Park, Seonghyuk Im et al.· arXiv.org· 2 citations
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