We prove $K_G^{\mathbb{C}}>1.35584631827168$ for the classical complex Grothendieck constant, closing more than one quarter of the gap between Davie's lower bound and Haagerup's upper bound. The numerical part of the proof is rigorously verified by interval arithmetic. The lower bound and the proof were discovered by the Odin Automatic AI Research Agent.
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 establish a new asymptotic lower bound for the volume of a maximal partial Room square of even order $n$. We prove that its volume is at least $\frac{2\sqrt{66}-6}{57}n^2-O(n)$, improving the leading coefficient $1/8$ in the lower bound obtained by Meszka and Rosa. The proof combines two counting arguments, one base...
For $f(T) = T^2 - 1$, we establish a lower bound for the relative dynamical house of wandering algebraic integers. This bound depends on the degree over the maximal abelian extension of the rationals. Our proof follows Dimitrov's approach to the Schinzel-Zassenhaus Conjecture and its generalization by Habegger-Schmidt.
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 show that there exists $c>0$ such that any subset of $\{1,\ldots,N\}$ having size $\gg N / \exp( (\log\log\log N)^c )$ contains a nontrivial pattern of the form $x,x+y,x+2y,x+y^3$. It is the first configuration of complexity strictly greater than $0$, other than refinements of arithmetic progressions, for which quan...
Bartłomiej Bychawski, Bartosz Głowacki, Ivan Spyrydonov· 0 citations
We prove an $\Omega(n^2)$ lower bound on the determinantal complexity of the power sum polynomial $\sum_{i=1}^n x_i^n$ over the field of complex numbers. A similar result was claimed in a recent paper of Sheshadri (arXiv:2606.13628), via an AI-assisted and AI-written proof. Assuming its correctness, this was the first...
Mrinal Kumar, Ben Lee Volk· 2 citations
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