I believe it means that the log is uniformly distributed, and the variable takes values in the range $[128, 4000]$.
From a footnote of the paper:
We will use the phrase drawn geometrically from A to B for 0 < A < B to mean drawing uniformly in the log domain between log(A) and log(B), exponentiating to get a number between A and B, and then rounding to the nearest integer. The phrase drawn exponentially means the same thing but without rounding.
x <- exp(runif(100000, log(128), log(4000)))
hist(x, breaks=100, xlim=c(128, 4000))
hist(log(x), breaks=100, xlim=c(log(128), log(4000)))