What does a tilde underneath an inequality mean?

I've recently come across an expression of the form $$\large x \lesssim y$$ What does this expression mean?

• Probably something like "asymptotically less than", but it's hard to know without a little more context... May 21 '14 at 20:58
• The symbols you probably mean are $\lesssim$ ($\mathtt{\backslash lesssim}$) and $\gtrsim$ ($\mathtt{\backslash gtrsim}$). I've never seen these used before but from a quick google search the first result just referenced them as "less/greater than approximately".
– Eff
May 21 '14 at 21:03
• @user112061 That was my first guess, but I just can't make sense out of what it means for something to be "approximately less than" something else. May 21 '14 at 21:05
• I have seen the expression $x \lesssim y$ used to mean that there exists a positive constant $C$ such that $x \leq C y$. It's convenient when one is making a series of estimates of that form and doesn't want to keep track of the constants. May 21 '14 at 21:06

In the context of partial differential equations and harmonic analysis, this notation is often used to mean

$$A \lesssim B \iff \exists C > 0 \text{ s.t. } A \leq C B$$

As written this expression is pretty damn useless when $A$ and $B$ are just two real numbers. What's more useful is supposing $A(\lambda), B(\lambda)$ are two families of objects parametrised by $\lambda\in \Lambda$ for some set. Then

$$A \lesssim B \iff \exists C > 0 \text{ s.t. } \forall \lambda\in \Lambda~,~ A(\lambda) \leq C B(\lambda)$$

Sometimes you will see when $A(\lambda,\pi), B(\lambda,\pi)$ are two families of objects parametrised by $(\lambda,\pi) \in \Lambda\times\Pi$, the notation

$$A\lesssim_\pi B \iff \forall \pi\in \Pi \exists C = C(\pi)>0 \text{ s.t. } \forall \lambda\in \Lambda ~,~ A(\lambda,\pi) \leq C(\pi) B(\lambda,\pi)$$

that is, the constant in the inequality is universal over $\lambda$ but may depend on $\pi$.

Similarly one sees the notation

$$A\approx B \iff A\lesssim B \text{ and } B\lesssim A$$

which also has the $\approx_\pi$ variant in the same way.

• I should remark that this notation is an extension of the big O notation which dates back over a century. May 22 '14 at 10:36
• How is it pronounced?
– user522521
Feb 21 '18 at 12:12
• In the context of PDE and harmonic analysis, usually we would read $A\lesssim B$ as "$A$ is bounded by $B$". Feb 21 '18 at 14:25
• Would you be so kind and point me to a reference (preferably a book) for the $A \lesssim_\pi B$ notation? Thank you. Apr 3 at 9:27
• @StefanPerko: see e.g. the Preface in Terence Tao, Nonlinear Dispersive Equations: Local and Global Analysis (bottom of page xiv). Apr 5 at 2:15

One possible interpretation of $x \lesssim y$ is, $$x < y \ \mbox{ or }\ x \approx y.$$

This is analogous to the way $x \le y$ signifies "$x < y$ or $x = y$."

But from the comments, clearly this is not the only way this symbol might be used. One would hope that prior to its first use in a publication, the same publication would define the symbol.

Maybe $x \lesssim y$ means $y-x<\epsilon$ fro some small $\epsilon$. In other words, the difference is small but x IS less than y

• This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post. - From Review Mar 23 '16 at 18:20
• @C.Dubussy This does answer the question and is exactly how I would interpret this kind of expression. Granted, that's from a physics perspective. In short, it's a mix of $\simeq$ or $\approx$ with $<$. Mar 23 '16 at 20:13
• This is not true if it is related to big O notation. If it is related to bigO, epsilon could be an integer constant less than infinity, albeit probably reasonably small. But a "small epsilon" implies a number which typically is much less than 1, but greater than zero. Jun 25 '18 at 14:53
• This is a quite common usage of the symbol in physics, when it makes sense to talk about two quantities being approximately equal, but at the same time knowing which one of them is the larger one. Oct 20 '18 at 17:55

The symbols are used frequently in high-energy physics by theorists and phenomenologists.

In that context, they mean less than about or greater than about, i.e., $$x \lesssim 10$$ means that $$x$$ is less than about 10, so as @DavidK suggests, either $$x<10$$ or $$x\approx 10$$.

E.g., there are experimental constraints on the mass of a hypothetical particle called the gluino. The limits are somewhat complicated, depending on many other assumptions. Rather than write something technical with many caveats and explanations, the situation is summarized by just saying $$m_\text{gluino} \gtrsim 1\,\text{TeV}$$, meaning that the gluino mass must be greater than about $$1\,\text{TeV}$$.

One of several possible meanings is a total preorder.