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Abstract A nonlinear version of Roth's theorem states that dense sets of integers contain configurations of the form , , . We obtain a multidimensional version of this result, which can be regarded as a first step toward effectivising those cases of the multidimensional polynomial Szemerédi theorem involving polynomials with distinct degrees. In addition, we prove an effective “popular” version of this result, showing that every dense set has some non‐zero such that the number of configurations with difference parameter is almost optimal. Perhaps surprisingly, the quantitative dependence in this result is exponential, compared to the tower‐type bounds encountered in the popular linear Roth theorem.
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On Waring's problem: Beyond Freĭman's theorem
Abstract Let satisfy . Freĭman's theorem shows that when , there exists such that all large integers are represented in the form , with , if and only if diverges. We make this theorem effective by showing that, for each fixed , it suffices to impose the conditionMore is established when the sequence of exponents forms an arithmetic progression. Thus, for example, when and , all large integers are represented in the form , with .
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- PAR ID:
- 10480764
- Publisher / Repository:
- Oxford University Press (OUP)
- Date Published:
- Journal Name:
- Journal of the London Mathematical Society
- Volume:
- 109
- Issue:
- 1
- ISSN:
- 0024-6107
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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