What do following asymptotic symbols mean? What do these symbols mean? I see them in analytic number theory.
$$\ll$$
$$\gg$$
$$\ll_\epsilon$$
$$\gg_\epsilon$$
$$\asymp$$
$$\sim$$
All these appear in here http://www.math.uiuc.edu/~lenfuchs/Binaryrep.pdf.
 A: These are all completely standard notations, and are used often in analytic number theory. (I've used them all at some point or another.) I would say that mathematicians working in analytic number theory would all be pretty comfortable with these notations and would not expect them to be defined in a paper. In a textbook, on the other hand, they'd probably be written down somewhere: for example, I believe there is a list of such definitions at the beginning of Montgomery and Vaughan's book.
We say that $f(x) \ll g(x)$ (implicitly understood as $x \to \infty$) if $f(x) = O(g(x))$, or equivalently if
\[\limsup_{x \to \infty} \frac{|f(x)|}{g(x)} = C < \infty\]
for some nonnegative constant $C$. Note that $g(x)$ is usually implicitly assumed to be a positive function, at least for all $x$ sufficiently large.
Similarly, $f(x) \gg g(x)$ (again implicitly understood as $x \to \infty$) if $g(x) = O(f(x))$, or equivalently if
\[\limsup_{x \to \infty} \frac{g(x)}{f(x)} = C < \infty\]
for some nonnegative constant $C$. Here both $f(x)$ and $g(x)$ are implicitly understood to be positive functions.
The notation $\ll_{\varepsilon}$ implies that the implicit constant $C$ depends on the parameter $\varepsilon$. For example, if we let $f(x) = \sum_{n \leq x} \mu(n)$, the Mertens function, and $g(x) = x^{1/2 + \varepsilon}$, which depends on the parameter $\varepsilon$, then the Riemann hypothesis is equivalent to the statement $f(x) \ll_{\varepsilon} g(x)$. This is also written as $f(x) = O_{\varepsilon}(g(x))$. This means that
\[\limsup_{x \to \infty} \frac{|f(x)|}{g(x)} = C < \infty\]
for some nonnegative constant $C = C(\varepsilon)$ that is allowed to depend on the parameter $\varepsilon$.
The notation $\gg_{\varepsilon}$ is defined analogously.
The notation $f(x) \asymp g(x)$ means that both $f(x) \gg g(x)$ and $f(x) \ll g(x)$ hold simultaneously. For an example, take $f(x) = x + \sin x$ and $g(x) = x$.
The notation $f(x) \sim g(x)$ is occasionally also written as $f(x) = (1 + o(1)) g(x)$ or $f(x) = g(x) + o(g(x))$. It simply means that $f(x) - g(x) = o(g(x))$, or equivalently that
\[\limsup_{x \to \infty} \frac{|f(x) - g(x)|}{g(x)} = 0.\]
The one other common definition that you don't mention is $\Omega$. We write $f(x) = \Omega(g(x))$ if
\[\limsup_{x \to \infty} \frac{|f(x)|}{g(x)}> 0.\]
It may be the case that this limit is $\infty$, and this is certainly allowed. For example, take $f(x) = x^2$ and $g(x) = x$. One can be more precise and write $f(x) = \Omega_+(g(x))$ if
\[\limsup_{x \to \infty} \frac{f(x)}{g(x)} > 0,\]
and similarly $f(x) = \Omega_-(g(x))$ if
\[\liminf_{x \to \infty} \frac{f(x)}{g(x)} < 0.\]
A: My guess would be the following:
$$ f \ll g \iff f \in \mathcal{O}(g) \iff \limsup_{X \to \infty} \left | \frac{ f}{g} \right | < \infty$$
The next I'm not sure, but I'd guess he means
$$ f \ll_{\epsilon} g(\epsilon) \iff \limsup_{X \to \infty} \left | \frac{ f}{g} \right | = M(\epsilon) < \infty $$
Then 
$$ f \asymp  g \iff f \ll g\quad  \&  \quad g \ll f$$ 
Lastly
$$ f \sim g \iff  \lim_{X \to \infty} \frac{f}{g} = 1 $$
A: The authors should have explained their notation, so shame on Bourgain and Fuchs.  I've never seen something like $\leq_{\epsilon}$.
