Abstract
We establish that, for almost all natural numbers N, there is a sum of two positive integral cubes lying in the interval \([N-N^{7/18+\varepsilon },N]\). Here, the exponent 7 / 18 lies half way between the trivial exponent 4 / 9 stemming from the greedy algorithm, and the exponent 1 / 3 constrained by the number of integers not exceeding X that can be represented as the sum of two positive integral cubes. We also provide analogous conclusions for sums of two positive integral k-th powers when \(k\ge 4\).
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Jörg Brüdern acknowledges support by Deutsche Forschungsgemeinschaft. Trevor D. Wooley is grateful for the support and excellent working conditions provided at Mathematisches Institut, Göttingen, through the Gauss Professorship of Akademie der Wissenschaften zu Göttingen, which greatly facilitated the preparation of this paper.
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Brüdern, J., Wooley, T.D. Additive representation in short intervals, II: sums of two like powers. Math. Z. 286, 179–196 (2017). https://doi.org/10.1007/s00209-016-1759-x
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DOI: https://doi.org/10.1007/s00209-016-1759-x