New Lower and Upper Bounds for the Grothendieck Constant
We establish new bounds on the Grothendieck constant $K_G$: \[
\frac{6π}{11}
\le
K_G
\le
\fracπ{2\log(1+\sqrt2)} - 10^{-4}. \] Methodologically, our lower bound approach differs from previous works by establishing limitations on the asymptotically optimal Krivine schemes, rather than giving explicit constructions of gap instances. Our upper bound is obtained by proposing and analyzing the first asymptotic construction of rounding schemes, whereas previous works only consider low-dimensional schemes. Together, these bounds determine the previously unknown tenths digit of $K_G$ to be $7$. The bounds were discovered by a long-running collaborative effort of humans and a long-horizon AI research system that we engineered.
\frac{6π}{11}
\le
K_G
\le
\fracπ{2\log(1+\sqrt2)} - 10^{-4}. \] Methodologically, our lower bound approach differs from previous works by establishing limitations on the asymptotically optimal Krivine schemes, rather than giving explicit constructions of gap instances. Our upper bound is obtained by proposing and analyzing the first asymptotic construction of rounding schemes, whereas previous works only consider low-dimensional schemes. Together, these bounds determine the previously unknown tenths digit of $K_G$ to be $7$. The bounds were discovered by a long-running collaborative effort of humans and a long-horizon AI research system that we engineered.
Summary
New research presents refined lower and upper bounds for the Grothendieck constant, offering a fresh perspective by focusing on limitations of Krivine schemes rather than traditional gap instances. This study also introduces an innovative asymptotic construction for rounding schemes, advancing the understanding of this mathematical constant.
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