I saw two less than signs on this Wikipedia article and I was wonder what they meant mathematically.
EDIT: It looks like this can use TeX commands. So I think this is the symbol: $\ll$
$\endgroup$ 18 Answers
$\begingroup$In the occurrence of "$\ll$" you are asking about, it means "much less than". If you look at the fourth entry here, this is the first meaning listed for $\ll$.
As Charles has correctly pointed out, this symbol is also used in advanced mathematics to describe a certain relationship in the growth of two functions. That is the second meaning listed.
$\endgroup$ 1 $\begingroup$"$a\ll b$" can also mean "$a$ at least as smaller than $b$ as it is needed for my arguments to be true".
It is in that sense that one sometimes writes, for example, "let $x$ be such that $0< x\ll 1$" to mean "let $x$ be a positive number as small as needed for the following to hold".
$\endgroup$ $\begingroup$It does not mean "much less than". It is the Vinogradov symbol, similar to the Hardy-Landau-etc. Big O notation.
$$f(x)\ll g(x)$$ means that there exists some $N$ and $k > 0$ such that, for all $x > N$, $f(x)<k\cdot g(x).$ In slightly more informal terms, it means that the asymptotic growth of $f(x)$ is no faster than that of $g(x)$.
$\endgroup$ 9 $\begingroup$This is the perfect example of the overloaded symbol. In measure theory, we use $\nu << \mu$ if the measure $\nu$ is absolutely continuous with respect to $\mu$, i.e., for any measurable $E$, we have $\mu(E) = 0\Rightarrow \nu(E) =0.$ Most mathematical symbols require a context to be interpreted unambiguously.
$\endgroup$ $\begingroup$Another way to think about the much less than $\ll$ is in the spirit of approx $\approx$. When you write $a \ll b$ you say that errors of size $a$ don't matter in assessing a quantity of size $b$. So in the article you reference, saying $k \ll N$ allows replacement of $N-k$ by $N$ to simplify the expression, or $N-k \approx N$. For practical purposes, such as the one in the article, an answer within $10\%$ is plenty good enough.
$\endgroup$ $\begingroup$Perhaps not its original intention, but we (my collaborators and former advisor) use $X \gg Y$ to mean that $X \geq c Y$ for a sufficiently large constant $c$. Precisely, we usually use it when we write things like:
$$ f(x) = g(x) + O(h(x)) \quad \Longrightarrow \quad f(x) = g(x) (1 + o(1)) $$
when $g(x) \gg h(x)$.
$\endgroup$ 4 $\begingroup$It means significantly smaller than, if I'm not mistaken.
$\endgroup$ $\begingroup$In programming the symbol $\ll$ means binary right shift. If we shift by one, that has the effect of multiplying a number by two. From this, the conclusion is quite obvious: $a \ll b$ means that $b$ is more that two times as large as $a$. In other words, $b$ is much larger than $a$.
$\endgroup$ 1
