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Mathematics

Counting up to four

Nature volume 397, pages 393–394 (1999)Cite this article

In 1936, Erdös and Turan1 stated this result as a conjecture — that is, something they believed was true but had no proof for. Some 40 years later it was proved for the first time by Szemerédi2. In the meantime, Roth3 had given a very interesting proof for the case k = 3. Roth's proof is remarkable in that it tells us the largest possible alignment, of given density δ, that does not contain three evenly spaced balls: the maximum number of marbles in such an alignment turns out to be exp{exp(C/δ)}, where C is a constant that does not depend on δ.

Now, a double exponential is certainly a very big number, but it pales in comparison with the much larger numbers that we get from Szemerédi's proof, and from subsequent ones, such as the remarkable proof by Furstenberg4, which relies on ergodic theory. Indeed, Roth's result was considered a notable achievement, and similar answers have long been sought for the case k > 3, in the hope that new estimates would lead to a better understanding of the combinations of numbers. This has just been achieved by Tim Gowers5, who proves that any alignment with density δ that does not contain four evenly spaced marbles must have fewer than exp{exp(1/δC)} marbles, where C is a constant which does not depend on δ. This is a breakthrough, because Gowers's method does not seem to be limited to k = 4; in fact, an unpublished manuscript by Gowers extends his result to all values of k, and gives a value of exp{exp(δ-exp {exp(k+10)})} as an upper bound for the number of marbles in an alignment with density δ,which does not contain k evenly spaced marbles.

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Figure 1: A game of marbles.

References

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  1. the University of Paris-Dauphine, Paris, 75775 Cedex 16, France

    Ivar Ekeland

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  1. Ivar Ekeland
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Ekeland, I. Counting up to four. Nature 397, 393–394 (1999). https://doi.org/10.1038/17026

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