Sign sequence

In mathematics, a sign sequence, or ±1–sequence or bipolar sequence, is a sequence of numbers, each of which is either 1 or −1. One example is the sequence (1, −1, 1, −1 ...).

Such sequences are commonly studied in discrepancy theory.

Erdős discrepancy problem

Around 1932 mathematician Paul Erdős conjectured that for any infinite ±1-sequence ${\displaystyle \textstyle \langle x_{1},x_{2},\ldots \rangle }$ and any integer C, there exist integers k and d such that:

$\left|\sum _{{i=1}}^{k}x_{{i\cdot d}}\right|>C$

The Erdős Discrepancy Problem asks for a proof or disproof of this conjecture.

In October 2010, this problem was taken up by the Polymath Project.^[1]

In February 2014, Alexei Lisitsa and Boris Konev of the University of Liverpool, UK,^[2] showed that every sequence of 1161 or more elements satisfies the conjecture in the special case C = 2, which proves the conjecture for C ≤ 2. This was the best such bound available at the time. Their proof relies on a SAT-solver computer algorithm whose output takes up 13 gigabytes of data, more than the entire text of Wikipedia at that time,^[3] so it cannot be verified by human mathematicians. However, human checking may not be necessary: if an independent computer verification returns the same results, the proof is likely to be correct.^[4]

In September 2015, Terence Tao announced a proof of the conjecture,^[5] building on work done in 2010 during Polymath5 (a form of crowdsourcing applied to mathematics)^[6] and a suggested link to the Elliott conjecture on pair correlations of multiplicative functions.^[7] His proof was published in 2016, as the first paper in the new journal Discrete Analysis.^[8]

Barker codes

A Barker code is a sequence of N values of +1 and −1,

${\displaystyle x_{j}{\text{ for }}j=1,\ldots ,N}$

such that

$\left|\sum _{{j=1}}^{{N-v}}x_{j}x_{{j+v}}\right|\leq 1\,$

for all $1\leq v<N$.^[9]

Barker codes of lengths 11 and 13 are used in direct-sequence spread spectrum and pulse compression radar systems because of their low autocorrelation properties.

See also

• Binary sequence
• Discrepancy of hypergraphs
• Rudin–Shapiro sequence

Notes

1. "The Erdős discrepancy problem". Polymath Project.
2. Konev, Boris; Lisitsa, Alexei (17 Feb 2014). "A SAT Attack on the Erdos Discrepancy Conjecture". arXiv:1402.2184.
3. see Wikipedia:Size of Wikipedia
4. Aron, Jacob (February 17, 2014). "Wikipedia-size maths proof too big for humans to check". New Scientist. Retrieved February 18, 2014.
5. Tao, Terence (2015-09-18). "The logarithmically averaged Chowla and Elliott conjectures for two-point correlations; the Erdos discrepancy problem". What's new.
6. article, New Scientist 30 Sep 15 retrieved 21.10.2015
7. article, New Scientist 25 Sep 15 retrieved 21.10.2015
8. Tao, Terence (2016). "The Erdős discrepancy problem". Discrete Analysis: 1–29. arXiv:1509.05363. doi:10.19086/da.609. ISSN 2397-3129. MR 3533300.
9. Barker, R. H. (1953). "Group Synchronizing of Binary Digital Sequences". Communication Theory. London: Butterworth. pp. 273–287.

References

• Chazelle, Bernard. The Discrepancy Method: Randomness and Complexity. Cambridge University Press. ISBN 0-521-77093-9.

External links

• The Erdős discrepancy problem – Polymath Project
• Computer cracks Erdős puzzle – but no human brain can check the answer—The Independent (Friday, 21 February 2014)