Suppose, I want to state that $a$ is less than $b$. However, I do not know $b$ exactly, but only that it is approximately $c$. With other words I want to state that $a$ is less than some value that is approximately $c$. I want to express this using one symbol, since I do not wish to introduce $b$ just for the purpose of making this statement. For example (using $∎$ as the desired symbol):
For $a ∎ 4.2$, the dynamics is chaotic.
Obvious symbols that come to mind for expressing this relation are $\lesssim$ and $\lessapprox$.
Is there any convention on which symbol to use for such a case or are there any good argument for or against either alternative regarding consistency and avoiding confusions? Related as well as an example of what kind of information I am looking for: I consider using $\sim$ instead of $\approx$ for approximately equal a bad idea, since it is used for mathematical equivalence as well as for proportional to (despite the existence of $\propto$) and thus it is often unclear what is meant by this symbol.
Best Answer
In Unicode, the symbol $\lessapprox$ is named “LESS-THAN OR APPROXIMATE”, which describes pretty much the meaning you are after. It also is built from the symbols $<$ (less than) and $\approx$ (approximately equal) in the very same way as $\leqq$ (less or equal) is built from $<$ (less) and $=$ (equal).
Given that commonly for "less than equal" a single line is used instead of a double line (i.e. $\leq$ instead of $\leqq$), using $\lesssim$ instead of $\lessapprox$ seems justified as well.
Indeed, in the comments to your question, Akiva Weinberger has confirmed that the latter symbols is actually used with that meaning in "NSA contexts" (which probably means non-standard analysis).
So in summary, I'd say for both symbols there's strong evidence that their use with that meaning is appropriate.
Another option might be to write $a<b+\epsilon$. Commonly $\epsilon$ is understood as a small positive quantity, so I'd argue that you are not really introducing a new variable there.