New bounds for equiangular lines and Balla's conjecture

Let $N_{\alpha }(d)$ denote the maximum number of equiangular lines in ${\mathbb{R}}^d$ with common angle $\arccos (\alpha )$. Balla conjectured that, if the spectral radius order ${\kappa }_{\frac{1-\alpha }{2\alpha }}$ of $\frac{1-\alpha }{2\alpha }$ is finite, then $$N_{\alpha }(d)\le \max \{\frac{(1-{\alpha }^2)(1-2{\alpha }^2)}{2{\alpha }^4},\lfloor \frac{{\kappa }_{\frac{1-\alpha }{2\alpha }}(d-1)}{{\kappa }_{\frac{1-\alpha }{2\alpha }}-1}\rfloor \},$$ for any $d\ge 1$. The conjecture has previously been verified only for $\alpha \in \{\frac{1}{3},\frac{1}{5},\frac{1}{1+2\sqrt{2}}\}$. In this paper, we prove that this conjecture holds for $\alpha =\frac{1}{1+2\sqrt{3}}$ and $\alpha =\sqrt{5}-2$. On the other hand, we show that Balla’s conjecture fails for infinitely many $\alpha$.