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).


closed as not constructive by Cheerful Parsnip, lhf, Grigory M, Asaf Karagila, Bill Cook Jan 12 '12 at 3:35

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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. $\endgroup$ – Graphth 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 $\endgroup$ – Graphth 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

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: $$\left|\sqrt\pi-\left(\frac{1}{10^5}\sum_{n=-\infty}^{\infty}e^{-n^2/10^{10}}\right)\right|\leq 10^{-4.2\cdot 10^{10}} $$

  • $\begingroup$ 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

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


(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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