Salem–Spencer set

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Short description: Progression-free set of numbers
For the set {1, 2, 4, 5, 10, 11, 13, 14}, all midpoints of two elements (the 28 yellow points) land outside the set, so no three elements can form an arithmetic progression

In mathematics, and in particular in arithmetic combinatorics, a Salem-Spencer set is a set of numbers no three of which form an arithmetic progression. Salem–Spencer sets are also called 3-AP-free sequences or progression-free sets. They have also been called non-averaging sets,[1][2] but this term has also been used to denote a set of integers none of which can be obtained as the average of any subset of the other numbers.[3] Salem-Spencer sets are named after Raphaël Salem and Donald C. Spencer, who showed in 1942 that Salem–Spencer sets can have nearly-linear size. However a later theorem of Klaus Roth shows that the size is always less than linear.

Examples

For [math]\displaystyle{ k=1,2,\dots }[/math] the smallest values of [math]\displaystyle{ n }[/math] such that the numbers from [math]\displaystyle{ 1 }[/math] to [math]\displaystyle{ n }[/math] have a [math]\displaystyle{ k }[/math]-element Salem-Spencer set are

1, 2, 4, 5, 9, 11, 13, 14, 20, 24, 26, 30, 32, 36, ... (sequence A065825 in the OEIS)

For instance, among the numbers from 1 to 14, the eight numbers

{1, 2, 4, 5, 10, 11, 13, 14}

form the unique largest Salem-Spencer set.[4]

This example is shifted by adding one to the elements of an infinite Salem–Spencer set, the Stanley sequence

0, 1, 3, 4, 9, 10, 12, 13, 27, 28, 30, 31, 36, 37, 39, 40, ... (sequence A005836 in the OEIS)

of numbers that, when written as a ternary number, use only the digits 0 and 1. This sequence is the lexicographically first infinite Salem–Spencer set.[5] Another infinite Salem–Spencer set is given by the cubes

0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, ... (sequence A000578 in the OEIS)

It is a theorem of Leonhard Euler that no three cubes are in arithmetic progression.[6]

Size

In 1942, Salem and Spencer published a proof that the integers in the range from [math]\displaystyle{ 1 }[/math] to [math]\displaystyle{ n }[/math] have large Salem–Spencer sets, of size [math]\displaystyle{ n/e^{O(\log n/\log\log n)} }[/math].[7] The denominator of this expression uses big O notation, and grows more slowly than any power of [math]\displaystyle{ n }[/math], so the sets found by Salem and Spencer have a size that is nearly linear. This bound disproved a conjecture of Paul Erdős and Pál Turán that the size of such a set could be at most [math]\displaystyle{ n^{1-\delta} }[/math] for some [math]\displaystyle{ \delta\gt 0 }[/math].[4][8] The construction of Salem and Spencer was improved by Felix Behrend in 1946, who found sets of size [math]\displaystyle{ n/e^{O(\sqrt{\log n})} }[/math].[9]

In 1952, Klaus Roth proved Roth's theorem establishing that the size of a Salem-Spencer set must be [math]\displaystyle{ O(n/\log\log n) }[/math].[10] Therefore, although the sets constructed by Salem, Spencer, and Behrend have sizes that are nearly linear, it is not possible to improve them and find sets whose size is actually linear. This result became a special case of Szemerédi's theorem on the density of sets of integers that avoid longer arithmetic progressions.[4] To distinguish Roth's bound on Salem–Spencer sets from Roth's theorem on Diophantine approximation of algebraic numbers, this result has been called Roth's theorem on arithmetic progressions.[11] After several additional improvements to Roth's theorem,[12][13][14][15] the size of a Salem–Spencer set has been proven to be [math]\displaystyle{ O\bigl(n(\log\log n)^4/\log n\bigr) }[/math].[16] An even better bound of [math]\displaystyle{ O\bigl(n/(\log n)^{1+\delta}\bigr) }[/math] (for some [math]\displaystyle{ \delta\gt 0 }[/math] that has not been explicitly computed) was announced in 2020 but has not yet been refereed and published.[17] In 2023 a new bound of [math]\displaystyle{ 2^{-O((\log N)^c)} \cdot N }[/math][18][19] was found and four days later the result was simplified with a little improvement to [math]\displaystyle{ \exp(-c(\log N)^{1/11})N }[/math],[20] these results have not yet been refereed and published either.

Construction

A simple construction for a Salem–Spencer set (of size considerably smaller than Behrend's bound) is to choose the ternary numbers that use only the digits 0 and 1, not 2. Such a set must be progression-free, because if two of its elements [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math] are the first and second members of an arithmetic progression, the third member must have the digit two at the position of the least significant digit where [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math] differ.[4] The illustration shows a set of this form, for the three-digit ternary numbers (shifted by one to make the smallest element 1 instead of 0).

Behrend's construction uses a similar idea, for a larger odd radix [math]\displaystyle{ 2d-1 }[/math]. His set consists of the numbers whose digits are restricted to the range from [math]\displaystyle{ 0 }[/math] to [math]\displaystyle{ d-1 }[/math] (so that addition of these numbers has no carries), with the extra constraint that the sum of the squares of the digits is some chosen value [math]\displaystyle{ k }[/math].[9] If the digits of each number are thought of as coordinates of a vector, this constraint describes a sphere in the resulting vector space, and by convexity the average of two distinct values on this sphere will be interior to the sphere rather than on it.[21] Therefore, if two elements of Behrend's set are the endpoints of an arithmetic progression, the middle value of the progression (their average) will not be in the set. Thus, the resulting set is progression-free.[9]

With a careful choice of [math]\displaystyle{ d }[/math], and a choice of [math]\displaystyle{ k }[/math] as the most frequently-occurring sum of squares of digits, Behrend achieves his bound.[9] In 1953, Leo Moser proved that there is a single infinite Salem–Spencer sequence achieving the same asymptotic density on every prefix as Behrend's construction.[1] By considering the convex hull of points inside a sphere, rather than the set of points on a sphere, it is possible to improve the construction by a factor of [math]\displaystyle{ \sqrt{\log n} }[/math].[21][22] However, this does not affect the size bound in the form stated above.

Generalization

The notion of Salem–Spencer sets (3-AP-free set) can be generalized to [math]\displaystyle{ k }[/math]-AP-free sets, in which [math]\displaystyle{ k }[/math] elements form an arithmetic progression if and only if they are all equal. (Rankin 1961) gave constructions of large [math]\displaystyle{ k }[/math]-AP-free sets.[23]

Computational results

Gasarch, Glenn, and Kruskal have performed a comparison of different computational methods for large subsets of [math]\displaystyle{ \{1,\dots n\} }[/math] with no arithmetic progression.[2] Using these methods they found the exact size of the largest such set for [math]\displaystyle{ n\le 187 }[/math]. Their results include several new bounds for different values of [math]\displaystyle{ n }[/math], found by branch-and-bound algorithms that use linear programming and problem-specific heuristics to bound the size that can be achieved in any branch of the search tree. One heuristic that they found to be particularly effective was the thirds method, in which two shifted copies of a Salem–Spencer set for [math]\displaystyle{ n }[/math] are placed in the first and last thirds of a set for [math]\displaystyle{ 3n }[/math].[2]

Applications

Script error: No such module "Chessboard". In connection with the Ruzsa–Szemerédi problem, Salem–Spencer sets have been used to construct dense graphs in which each edge belongs to a unique triangle.[24]

Salem–Spencer sets have also been used in theoretical computer science. They have been used in the design of the Coppersmith–Winograd algorithm for fast matrix multiplication,[25] and in the construction of efficient non-interactive zero-knowledge proofs.[26] Recently, they have been used to show size lower bounds for graph spanners,[27] and the strong exponential time hypothesis based hardness of the subset sum problem.[28]

These sets can also be applied in recreational mathematics to a mathematical chess problem of placing as few queens as possible on the main diagonal of an [math]\displaystyle{ n\times n }[/math] chessboard so that all squares of the board are attacked. The set of diagonal squares that remain unoccupied must form a Salem–Spencer set, in which all values have the same parity (all odd or all even). The smallest possible set of queens is the complement of the largest Salem–Spencer subset of the odd numbers in [math]\displaystyle{ \{1,\dots n\} }[/math]. This Salem-Spencer subset can be found by doubling and subtracting one from the values in a Salem–Spencer subset of all the numbers in [math]\displaystyle{ \{1,\dots n/2\}. }[/math][29]

References

  1. ↑ 1.0 1.1 "On non-averaging sets of integers", Canadian Journal of Mathematics 5: 245–252, 1953, doi:10.4153/cjm-1953-027-0 
  2. ↑ 2.0 2.1 2.2 "Finding large 3-free sets. I. The small n case", Journal of Computer and System Sciences 74 (4): 628–655, 2008, doi:10.1016/j.jcss.2007.06.002, https://www.cs.umd.edu/~gasarch/papers/3apI.pdf 
  3. ↑ Abbott, H. L. (1976), "On a conjecture of Erdős and Straus on non-averaging sets of integers", Proceedings of the Fifth British Combinatorial Conference (Univ. Aberdeen, Aberdeen, 1975), Congressus Numerantium, XV, Winnipeg, Manitoba: Utilitas Math., pp. 1–4 
  4. ↑ 4.0 4.1 4.2 4.3 Dybizbański, Janusz (2012), "Sequences containing no 3-term arithmetic progressions", Electronic Journal of Combinatorics 19 (2): P15:1–P15:5, doi:10.37236/2061, https://www.combinatorics.org/ojs/index.php/eljc/article/view/v19i2p15 
  5. ↑ Sloane, N. J. A., ed. "Sequence A005836". OEIS Foundation. https://oeis.org/A005836. 
  6. ↑ "Greedy algorithm, arithmetic progressions, subset sums and divisibility", Discrete Mathematics 200 (1–3): 119–135, 1999, doi:10.1016/S0012-365X(98)00385-9 
  7. ↑ "On Sets of Integers Which Contain No Three Terms in Arithmetical Progression", Proceedings of the National Academy of Sciences 28 (12): 561–563, December 1942, doi:10.1073/pnas.28.12.561, PMID 16588588, Bibcode: 1942PNAS...28..561S 
  8. ↑ "On some sequences of integers", Journal of the London Mathematical Society 11 (4): 261–264, 1936, doi:10.1112/jlms/s1-11.4.261, https://users.renyi.hu/~p_erdos/1936-05.pdf 
  9. ↑ 9.0 9.1 9.2 9.3 "On sets of integers which contain no three terms in arithmetical progression", Proceedings of the National Academy of Sciences 32 (12): 331–332, December 1946, doi:10.1073/pnas.32.12.331, PMID 16578230, Bibcode: 1946PNAS...32..331B 
  10. ↑ "Sur quelques ensembles d'entiers", Comptes rendus de l'Académie des Sciences 234: 388–390, 1952 
  11. ↑ Bloom, Thomas; Sisask, Olaf (2019), "Logarithmic bounds for Roth's theorem via almost-periodicity", Discrete Analysis 2019 (4), doi:10.19086/da.7884 
  12. ↑ "Integer sets containing no arithmetic progressions", Journal of the London Mathematical Society, Second Series 35 (3): 385–394, 1987, doi:10.1112/jlms/s2-35.3.385 
  13. ↑ "Integer sets containing no arithmetic progressions", Acta Mathematica Hungarica 56 (1–2): 155–158, 1990, doi:10.1007/BF01903717 
  14. ↑ "On triples in arithmetic progression", Geometric and Functional Analysis 9 (5): 968–984, 1999, doi:10.1007/s000390050105 
  15. ↑ "On Roth's theorem on progressions", Annals of Mathematics, Second Series 174 (1): 619–636, 2011, doi:10.4007/annals.2011.174.1.20 
  16. ↑ Bloom, T. F. (2016), "A quantitative improvement for Roth's theorem on arithmetic progressions", Journal of the London Mathematical Society, Second Series 93 (3): 643–663, doi:10.1112/jlms/jdw010 
  17. ↑ Bloom, Thomas; Sisask, Olaf (2020), Breaking the logarithmic barrier in Roth's theorem on arithmetic progressions ; see also "To cheer you up in difficult times 7: Bloom and Sisask just broke the logarithm barrier for Roth's theorem!", Combinatorics and more, July 8, 2020, https://gilkalai.wordpress.com/2020/07/08/to-cheer-you-up-in-difficult-times-7-bloom-and-sisask-just-broke-the-logarithm-barrier-for-roths-theorem/ 
  18. ↑ Kelley, Zander; Meka, Raghu (2023-02-10). "Strong Bounds for 3-Progressions". arXiv:2302.05537 [math.NT].
  19. ↑ Sloman, Leila (2023-03-21). "Surprise Computer Science Proof Stuns Mathematicians" (in en). https://www.quantamagazine.org/surprise-computer-science-proof-stuns-mathematicians-20230321/. 
  20. ↑ Bloom, Thomas F.; Sisask, Olof (2023-02-14). "The Kelley--Meka bounds for sets free of three-term arithmetic progressions". arXiv:2302.07211 [math.NT].
  21. ↑ 21.0 21.1 Elkin, Michael (2011), "An improved construction of progression-free sets", Israel Journal of Mathematics 184: 93–128, doi:10.1007/s11856-011-0061-1 
  22. ↑ "A note on Elkin's improvement of Behrend's construction", Additive number theory: Festschrift in honor of the sixtieth birthday of Melvyn B. Nathanson, New York: Springer, 2010, pp. 141–144, doi:10.1007/978-0-387-68361-4_9 
  23. ↑ "XXIV: Sets of integers containing not more than a given number of terms in arithmetical progression", Proceedings of the Royal Society of Edinburgh, Section A: Mathematical and Physical Sciences 65 (4): 332–344, 1961, doi:10.1017/S0080454100017726 
  24. ↑ "Triple systems with no six points carrying three triangles", Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), Vol. II, Colloq. Math. Soc. János Bolyai, 18, Amsterdam and New York: North-Holland, 1978, pp. 939–945 
  25. ↑ "Matrix multiplication via arithmetic progressions", Journal of Symbolic Computation 9 (3): 251–280, 1990, doi:10.1016/S0747-7171(08)80013-2 
  26. ↑ Lipmaa, Helger (2012), "Progression-free sets and sublinear pairing-based non-interactive zero-knowledge arguments", in Cramer, Ronald, Theory of Cryptography: 9th Theory of Cryptography Conference, TCC 2012, Taormina, Sicily, Italy, March 19–21, 2012, Proceedings, Lecture Notes in Computer Science, 7194, Springer, pp. 169–189, doi:10.1007/978-3-642-28914-9_10, http://eprint.iacr.org/2011/009 
  27. ↑ Abboud, Amir; Bodwin, Greg (2017), "The 4/3 additive spanner exponent is tight", Journal of the ACM 64 (4): A28:1–A28:20, doi:10.1145/3088511 
  28. ↑ Abboud, Amir (2019), "SETH-based lower bounds for subset sum and bicriteria path", Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, San Diego, California, USA, January 6-9, 2019, Society for Industrial and Applied Mathematics, pp. 41–57, doi:10.1137/1.9781611975482.3 
  29. ↑ Cockayne, E. J.; Hedetniemi, S. T. (1986), "On the diagonal queens domination problem", Journal of Combinatorial Theory, Series A 42 (1): 137–139, doi:10.1016/0097-3165(86)90012-9 

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