Erdős–Fuchs theorem

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In mathematics, in the area of additive number theory, the Erdős–Fuchs theorem is a statement about the number of ways that numbers can be represented as a sum of elements of a given additive basis, stating that the average order of this number cannot be too close to being a linear function.

The theorem is named after Paul Erdős and Wolfgang Heinrich Johannes Fuchs, who published it in 1956.[1]

Statement

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Let   be an infinite subset of the natural numbers and   its representation function, which denotes the number of ways that a natural number   can be expressed as the sum of   elements of   (taking order into account). We then consider the accumulated representation function   which counts (also taking order into account) the number of solutions to  , where  . The theorem then states that, for any given  , the relation   cannot be satisfied; that is, there is no   satisfying the above estimate.

Theorems of Erdős–Fuchs type

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The Erdős–Fuchs theorem has an interesting history of precedents and generalizations. In 1915, it was already known by G. H. Hardy[2] that in the case of the sequence   of perfect squares one has   This estimate is a little better than that described by Erdős–Fuchs, but at the cost of a slight loss of precision, P. Erdős and W. H. J. Fuchs achieved complete generality in their result (at least for the case  ). Another reason this result is so celebrated may be due to the fact that, in 1941, P. Erdős and P. Turán[3] conjectured that, subject to the same hypotheses as in the theorem stated, the relation   could not hold. This fact remained unproven until 1956, when Erdős and Fuchs obtained their theorem, which is even stronger than the previously conjectured estimate.

Improved versions for h = 2

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This theorem has been extended in a number of different directions. In 1980, A. Sárközy[4] considered two sequences which are "near" in some sense. He proved the following:

  • Theorem (Sárközy, 1980). If   and   are two infinite subsets of natural numbers with  , then   cannot hold for any constant  .

In 1990, H. L. Montgomery and R. C. Vaughan[5] were able to remove the log from the right-hand side of Erdős–Fuchs original statement, showing that   cannot hold. In 2004, Gábor Horváth[6] extended both these results, proving the following:

  • Theorem (Horváth, 2004). If   and   are infinite subsets of natural numbers with   and  , then   cannot hold for any constant  .

General case (h ≥ 2)

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The natural generalization to Erdős–Fuchs theorem, namely for  , is known to hold with same strength as the Montgomery–Vaughan's version. In fact, M. Tang[7] showed in 2009 that, in the same conditions as in the original statement of Erdős–Fuchs, for every   the relation   cannot hold. In another direction, in 2002, Gábor Horváth[8] gave a precise generalization of Sárközy's 1980 result, showing that

  • Theorem (Horváth, 2002) If   ( ) are   (at least two) infinite subsets of natural numbers and the following estimates are valid:
  1.  
  2.   (for  )
then the relation:
 
cannot hold for any constant  .

Non-linear approximations

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Yet another direction in which the Erdős–Fuchs theorem can be improved is by considering approximations to   other than   for some  . In 1963, Paul T. Bateman, Eugene E. Kohlbecker and Jack P. Tull[9] proved a slightly stronger version of the following:

  • Theorem (Bateman–Kohlbecker–Tull, 1963). Let   be a slowly varying function which is either convex or concave from some point onward. Then, on the same conditions as in the original Erdős–Fuchs theorem, we cannot have  , where   if   is bounded, and   otherwise.

At the end of their paper, it is also remarked that it is possible to extend their method to obtain results considering   with  , but such results are deemed as not sufficiently definitive.

See also

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  • Erdős–Tetali theorem: For any  , there is a set   which satisfies  . (Existence of economical bases)
  • Erdős–Turán conjecture on additive bases: If   is an additive basis of order 2, then  . (Bases cannot be too economical)

References

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  1. ^ Erdős, P.; Fuchs, W. H. J. (1956). "On a Problem of Additive Number Theory". Journal of the London Mathematical Society. 31 (1): 67–73. doi:10.1112/jlms/s1-31.1.67. hdl:2027/mdp.39015095244037.
  2. ^ Hardy, G. H. (1915). "On the expression of a number as the sum of two squares". Quarterly Journal of Mathematics. 46: 263–83.
  3. ^ Erdős, P.; Turán, P. (1941). "On a problem of Sidon in additive number theory, and some related problems". Journal of the London Mathematical Society. Series 1. 16 (4): 212–215. doi:10.1112/jlms/s1-16.4.212.
  4. ^ Sárközy, András (1980). "On a theorem of Erdős and Fuchs". Acta Arithmetica. 37: 333–338. doi:10.4064/aa-37-1-333-338.
  5. ^ Montgomery, H. L.; Vaughan, R. C. (1990). "On the Erdős–Fuchs theorem". In Baker, A; Bollobas, B; Hajnal, A (eds.). A tribute to Paul Erdős. Cambridge University Press. pp. 331–338. doi:10.1017/CBO9780511983917.025. ISBN 9780511983917.
  6. ^ Horváth, G. (2004). "An improvement of an extension of a theorem of Erdős and Fuchs". Acta Mathematica Hungarica. 104: 27–37. doi:10.1023/B:AMHU.0000034360.41926.5a.
  7. ^ Tang, Min (2009). "On a generalization of a theorem of Erdős and Fuchs". Discrete Mathematics. 309 (21): 6288–6293. doi:10.1016/j.disc.2009.07.006.
  8. ^ Horváth, Gábor (2002). "On a theorem of Erdős and Fuchs". Acta Arithmetica. 103 (4): 321–328. Bibcode:2002AcAri.103..321H. doi:10.4064/aa103-4-2.
  9. ^ Bateman, Paul T.; Kohlbecker, Eugene E.; Tull, Jack P. (1963). "On a theorem of Erdős and Fuchs in additive number theory". Proceedings of the American Mathematical Society. 14 (2): 278–284. doi:10.1090/S0002-9939-1963-0144876-1.

Further reading

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