Abstract
According to the Erdős discrepancy conjecture, for any infinite ±1 sequence, there exists a homogeneous arithmetic progression of unbounded discrepancy. In other words, for any ±1 sequence (x 1,x 2,...) and a discrepancy C, there exist integers m and d such that \(|\sum_{i=1}^m x_{i \cdot d}| > C\). This is an 80-year-old open problem and recent development proved that this conjecture is true for discrepancies up to 2. Paul Erdős also conjectured that this property of unbounded discrepancy even holds for the restricted case of completely multiplicative sequences, namely sequences (x 1,x 2,...) where x a ·b = x a ·x b for any a,b ≥ 1. The longest such sequence of discrepancy 2 has been proven to be of size 246. In this paper, we prove that any completely multiplicative sequence of size 127,646 or more has discrepancy at least 4, proving the Erdős discrepancy conjecture for discrepancy up to 3. In addition, we prove that this bound is tight and increases the size of the longest known sequence of discrepancy 3 from 17,000 to 127,645. Finally, we provide inductive construction rules as well as streamlining methods to improve the lower bounds for sequences of higher discrepancies.
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References
Beck, J., Fiala, T.: Integer-Making theorems. Discrete Applied Mathematics 3, 1–8 (1981)
Spencer, J.: Six standard deviations suffice. Transactions of the American Mathematical Society 289, 679–706 (1985)
Erdös, P.: Some of my favourite problems which recently have been solved. North-Holland Mathematics Studies 74, 59–79 (1982)
Nikolov, A., Talwar, K.: On the hereditary discrepancy of homogeneous arithmetic progressions. arXiv preprint arXiv:1309.6034 (2013)
Konev, B., Lisitsa, A.: A sat attack on the erdos discrepancy conjecture (2014)
Heule, M.J., Hunt, W.A., Wetzler, N.: Trimming while checking clausal proofs. In: Formal Methods in Computer-Aided Design (FMCAD), pp. 181–188. IEEE (2013)
Biere, A.: Lingeling, plingeling and treengeling entering the sat competition 2013. In: Proceedings of SAT Competition (2013); Solver and (2013) 51
Audemard, G., Simon, L.: Glucose 2.3 in the sat 2013 competition. In: Proceedings of SAT Competition (2013); Solver and (2013) 42
Gomes, C., Sellmann, M.: Streamlined constraint reasoning. In: Wallace, M. (ed.) CP 2004. LNCS, vol. 3258, pp. 274–289. Springer, Heidelberg (2004)
Kouril, M., Franco, J.: Resolution tunnels for improved sat solver performance. In: Bacchus, F., Walsh, T. (eds.) SAT 2005. LNCS, vol. 3569, pp. 143–157. Springer, Heidelberg (2005)
Le Bras, R., Gomes, C.P., Selman, B.: From streamlined combinatorial search to efficient constructive procedures. In: Proceedings of the 15th International Conference on Artificial Intelligence, AAAI 2012 (2012)
Le Bras, R., Gomes, C.P., Selman, B.: Double-wheel graphs are graceful. In: Proceedings of the Twenty-Third International Joint Conference on Artificial Intelligence, IJCAI 2013, pp. 587–593. AAAI Press (2013)
Polymath1: Matryoshka Sequences, http://michaelnielsen.org/polymath1/index.php?title=Matryoshka_Sequences (accessed April 11, 2014)
Wetzler, N., Heule, M.J., Hunt Jr., W.A.: (Drat-trim: Efficient checking and trimming using expressive clausal proofs)
Konev, B., Lisitsa, A.: Computer-aided proof of erdos discrepancy properties. arXiv preprint arXiv:1405.3097 (2014)
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Le Bras, R., Gomes, C.P., Selman, B. (2014). On the Erdős Discrepancy Problem. In: O’Sullivan, B. (eds) Principles and Practice of Constraint Programming. CP 2014. Lecture Notes in Computer Science, vol 8656. Springer, Cham. https://doi.org/10.1007/978-3-319-10428-7_33
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DOI: https://doi.org/10.1007/978-3-319-10428-7_33
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