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单词 ErdHosTuranConjecture
释义

Erdős-Turan conjecture


Erdős-Turan conjecture asserts there exist no asymptotic basis (http://planetmath.org/AsymptoticBasis) A0 of order 2 such that its representation function

rA,2(n)=a1+a2=na1a21

is boundedPlanetmathPlanetmath.

Alternatively, the question can be phrased as whether there exists a power seriesMathworldPlanetmath F with coefficients 0 and 1 such that all coefficients of F2 are greater than 0, but are bounded.

If we replace set of nonnegative integers by the set of all integers, then the question was settled by Nathanson[2] in negative, that is, there exists a set A such that rA,2(n)=1.

References

  • 1 Heini Halberstam and Klaus Friedrich Roth. Sequences. Springer-Verlag, second edition, 1983. http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0498.10001Zbl 0498.10001.
  • 2 Melvyn B. Nathanson. Every functionMathworldPlanetmath is the representation function of an additive basisfor the integers. http://front.math.ucdavis.edu/math.NT/0302091arXiv:math.NT/0302091.
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更新时间:2025/5/4 1:09:35