A Discrepancy Problem: Balancing Infinite Dimensional Vectors



As a corollary of a general balancing result, we prove that there exists a balanced “2-coloring” g of the set of natural numbers \(\mathbb{N}\) such that simultaneously for all integers d ≥ 1, every (finite) arithmetic progression of difference d has discrepancy D g (d) ≤ d 8+ɛ , independently of the starting point and the length of the arithmetic progression. Formally, for every ɛ > 0 there exists a function \(g: \mathbb{N} \rightarrow \{-1,1\}\) such that
$$\displaystyle{ D_{g}(d) =\max _{a\geq 1,m\geq 1}\left \vert \sum _{i=0}^{m-1}g(a + id)\right \vert \leq d^{8+\varepsilon } }$$
for all sufficiently large dd 0(ɛ). This reduces an old superexponential upper bound ≤ d! of Cantor, Erdős, Schreiber, and Straus to a polynomial upper bound. Note that the polynomial range is the correct range, since a well known result of Roth implies the lower bound \(D_{g}(d) \geq \sqrt{d}/20\) for every \(g: \mathbb{N} \rightarrow \{-1,1\}\).We derive this concrete number theoretic upper bound result about arithmetic progressions from a very general vector balancing result. It is about balancing infinite dimensional vectors in the maximum norm, and it is interesting in its own right (possibly, more interesting than the special case above).



I am very grateful to I. Bárány for his remarks and suggestions, and to D. Reimer for his help in formulating the proof of Theorem 2.


  1. 1.
    J. Beck, Balancing families of integer sequences. Combinatorica 1, 209–216 (1981)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    J. Beck, Roth’s estimate of discrepancy of integer sequences is nearly sharp. Combinatorica 1, 319–325 (1981)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    J. Beck, V.T. Sós, Discrepancy theory, Chap. 26, in Handbook of Combinatorics, ed. by R. Graham, M. Grőtschel, L. Lovász (Elsevier, Amsterdam, 1995), pp. 1405–1446Google Scholar
  4. 4.
    J. Beck, J. Spencer, Well-distributed 2-colorings of integers relative to long arithmetic progressions. Acta Arith. 43, 287–294 (1984)MathSciNetMATHGoogle Scholar
  5. 5.
    P. Erdős, Extremal problems in number theory II (in Hungarian). Mat. Lapok 17, 135–155 (1966)MathSciNetMATHGoogle Scholar
  6. 6.
    P. Erdős, Problems and results on combinatorial number theory, in A Survey of Combinatorial Theory, ed. by J.N. Srivastava, et al. (North-Holland, Amsterdam, 1973), pp. 117–138CrossRefGoogle Scholar
  7. 7.
    J. Matoušek, J. Spencer, Discrepancy in arithmetic progressions. J. Am. Math. Soc. 9(1), 195–204 (1996)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    K.F. Roth, Remark concerning integer sequences. Acta Arith. 9, 257–260 (1964)MathSciNetMATHGoogle Scholar
  9. 9.
    T. Tao, The Erdős discrepancy problem. arXiv: 1509.05363v5, see also the new journal Discrete AnalysisGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsRutgers UniversityPiscatawayUSA

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