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You can read a lot about what large number in mathematics are, like Skewes' number, Moser's number or even Graham's number.

So just for the sake of non-discrimination, I ask, what is the smallest constant $\epsilon$ that has explicitly appeared in a published paper?

Comment: As usual let's assume $\epsilon >0$! And only numbers in the domain of real numbers are allowed (Thx to Holowitz).

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    $\begingroup$ $\epsilon$, usually by definition it stands for any number greater than 0, so it represents numbers arbitrarily close to 0. $\endgroup$ – GeoffDS Jan 11 '12 at 17:01
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    $\begingroup$ @Graphth: this is similar to saying that the largest number used in a proof is $N$, which is often allowed to tend to infinity. Neither $\epsilon$ nor $N$ stand for particular numbers; but particular numbers, I believe, are what Andreas is asking for. $\endgroup$ – Niel de Beaudrap Jan 11 '12 at 17:05
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    $\begingroup$ @NieldeBeaudrap It's a joke $\endgroup$ – GeoffDS Jan 11 '12 at 17:15
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    $\begingroup$ I think this question is currently too vague. A better formulation might be: "What is the smallest constant that has explicitly appeared in a published paper?" $\endgroup$ – Cheerful Parsnip Jan 11 '12 at 18:06
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    $\begingroup$ From Littlewood's "A Mathematician's Miscellany", p.38: "A minute I wrote (about 1917) for the Ballistic Office ended with the sentence 'Thus $\sigma$ should be made as small as possible'. This did not appear in the printed minute. But P. J. Grigg said, 'what is that?' A speck in a blank space at the end proved to be the tiniest $\sigma$ I have ever seen (the printers must have scoured London for it)." $\endgroup$ – Per Manne Jul 11 '14 at 13:13
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The article "Strange Series and High Precision Fraud", J. M. Borwein and P. B. Borwein, The American Mathematical Monthly, Vol. 99, No. 7, (Aug-Sep, 1992), pp. 622-640, contains some interesting small error terms. In particular on p.639: $$\left|\sqrt\pi-\left(\frac{1}{10^5}\sum_{n=-\infty}^{\infty}e^{-n^2/10^{10}}\right)\right|\leq 10^{-4.2\cdot 10^{10}} $$

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  • $\begingroup$ Seems a bit like cheating with all the $10^x$'s, but ok. We are getting closer... $\endgroup$ – draks ... Jan 11 '12 at 20:15
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Here's a case where $10^{-2576}$ came up.

In 1928, A. S. Besicovitch introduced and studied regular and irregular $1$-sets in the plane (a fundamental notion in fractal geometry). He proved that the lower $1$-density of a regular $1$-set $E$ in the plane is equal to $1$ (the maximum possible value) at almost all (Lebesgue measure) points in $E$. To show how differently irregular $1$-sets in the plane behaved, Besicovitch proved that the lower $1$-density of an irregular $1$-set $F$ in the plane is bounded below $1$ at almost all (Lebesgue measure) points in $F$. The bound that Besicovitch obtained in 1928 was $1 - 10^{-2576}$.

In 1934, Besicovitch managed to improve this to $\frac{3}{4}$. [See Section 3.3 of Kenneth J. Falconer, The Geometry of Fractal Sets, Cambridge University Press, 1985.] "Besicovitch's $\frac{1}{2}$-problem" (presently unsolved, I believe) is to find the best almost everywhere upper bound for irregular sets. Besicovitch himself showed by a specific example that this bound is at least $\frac{1}{2},$ and he conjectured that it is equal to $\frac{1}{2}.$ In 1992, David Preiss and Jaroslav Tiser proved the bound is at most $\frac{2 \;+ \;\sqrt{46}}{12},$ which is approximately $0.73186.$ For a lot more about the Besicovitch $\frac{1}{2}$-problem, see Hany M. Farag's papers at

http://www.math.caltech.edu/people/farag.html

(50 minutes later)

Here's more about the first Besicovitch reference:

On p. 454 of the paper below, about $\frac{2}{3}$ down the page, is the following (italics in the original). [Note: The paper is freely available on the Internet.]

At almost all points of an irregular set the lower density is less than $1 - 10^{-2576}.$

Abram Samoilovitch Besicovitch (1891-1970), On the fundamental geometrical properties of linearly measurable plane sets of points, Mathematische Annalen 98 (1928), 422-464.

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