I am reading materials about Lenstra–Lenstra–Lovász lattice basis reduction algorithm on Wikipedia.
What is the definition of this notation $\lfloor x\rceil$ appearing in the description of the algorithm?
From the context, I guess $x=\lfloor x\rfloor$ if $\{x\}<\frac{1}{2}$, and $x=\lceil x \rceil$ if $\{x\}\ge\frac{1}{2}$.
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It is the nearest integer function. I've seen the notation show up in some cryptography papers.
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The notation is first mentioned on page 381 of the book An Introduction to Mathematical Cryptography, per the linked Wikipedia article, where it is applied to scalars (real numbers) and signifies "round to nearest integer" in connection with Babai's algorithm:
L. Babai. On Lovász’ lattice reduction and the nearest lattice point problem. Combinatorica, 6(1):1–13, 1986
Subsequent mentions within that text apply it to real polynomials with the meaning of rounding coefficients to the nearest integers (see top of page 451) and to real vectors with the obvious componentwise application (see bottom of page 451). The former explanation is repeated within a text box on page 457, and the latter usage is illustrated in a proof just following that.
It seems curious to omit explanation for the use of this notation in the "figure" on page 411, from which the LLL lattice reduction algorithm was cribbed by the Wikipedia article. Even more curious is the omission of this from the List of Notation postscript that begins on page 489.
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For what it's worth, the existence of a "best integer approximation" amounts to the fact that $\mathbb{Z} \subseteq \mathbb{R}$ is "weakly Chebyshev," by which I mean that for all $x \in \mathbb{R}$, there exists $y \in \mathbb{Z}$ such that for all $y' \in \mathbb{Z}$, we have $d(x,y) \leq d(x,y')$. Weakly chebyshev sets make sense in any metric space. If we augment the condition that there exists such a $y$ with a uniqueness stipulation, we get the notion of a Chebyshev set. It turns out that the Chebyshev subsets of the real line are precisely $$[a,b], \quad (-\infty,b], \quad [a,\infty), \quad (-\infty,\infty),$$
see here for a proof.
It looks like a combination of ceiling and floor, so my guess is the notation could be reasonably used to denote rounding to the nearest integer.