What is the meaning of this symbol $\ll_d$?

Question

I apologize for the simple question but I'm reading a paper "On the Convex Hull Of The Integer Points In A Disc" and I'm confused by some notation. They say $$\# \textrm{ vertices of }P \ll_{\;d} (vol P)^{\frac{d-1}{d+1}}$$ And I'm unfamiliar with the meaning of $\ll_{\;d}$. Here, $P \subset R^d$ is a convex polytope with integral vertices and nonempty interior.

Answer 1

It means that the left side is bounded by a constant $C_{d} > 0$ times the right side, and the constant $C_{d}$ depends only on the number $d$ and nothing else.

Sometimes this is written instead with "big O" notation $x= O_{d}(y)$ (or, equivalently, $x\leq O_{d}(y)$).

** Added later by request: A source for this notation is the wikipedia article:

https://en.wikipedia.org/wiki/Big_O_notation#History_(Bachmann%E2%80%93Landau,_Hardy,_and_Vinogradov_notations)

The link talks about the meaning of $\ll$ as the same thing as big O. This is also called "Vinogradov's notation". It's common practice to put subscripts if the implicit constant in the big O depends on some parameters.

It's probably good to caution that sometimes people interpret $\ll$ as "much much less than" which might make one think of "little o" notation. So maybe it's always good to read carefully. But the article in the question explicitly says Vinogradov's notation.

Here's another good article which talks about these issues:

https://faculty.math.illinois.edu/~hildebr/595ama/ama-ch2.pdf

Sections 2.1.5, 2.1.6, and 2.1.7 are relevant to everything I've said.