# What does “hold uniformly” mean in the context of asymptotic analysis?

What does "hold uniformly" mean in the statement of Theorem 1.7 in A Faster Subquadratic Algorithm for Finding Outlier Correlations? Here's the theorem text ("hold uniformly" is in the last line):

• I don't find it very clear in this context. Possibly what they mean is that the constants and log factors hidden in the $\tilde{O}$ do not change even if $n,\tau,\rho$ change. – usul Jul 19 '18 at 17:16

For example, $f_n(x)= \sin(nx)$ is uniformly bounded by $1$, while $g_n(x)=n\cos(nx)$ is not. In fact, $g_n$ isn’t uniformly bounded by anything, as any bound on it needs to depend on $n$.