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On Waring’s problem: Two cubes and seven biquadrates. (English) Zbl 1003.11045

For natural numbers \(n\), let \(\nu(n)\) denote the number of representations of \(n\) as the sum of two positive integral cubes and seven biquadrates. The authors prove that \(\nu(n)\gg n^{43/36}(\log n)^{-1}\) for all large \(n\). The methods of this paper are of sufficient power to establish a lower bound of the form \(\nu(n)\gg n^{43/36+\delta}\), for large \(n\), with some small positive number \(\delta\). It is conjectured that \(\nu(n)\) is as large as \(n^{17/12}\).
The proof is constructed on various advanced technologies concerning the Hardy-Littlewood method, and especially requires the “breaking classical convexity” device of T. D. Wooley [Invent. Math. 122, 421-451 (1995; Zbl 0851.11055)] and an efficient differencing process restricted to minor arcs which originates in Vaughan’s work on Waring’s problem for cubes. Sharp mean value estimates for biquadratic smooth Weyl sums are established by applying Wooley’s device, and they are of independent importance.

MSC:

11P05 Waring’s problem and variants
11P55 Applications of the Hardy-Littlewood method
11L07 Estimates on exponential sums

Citations:

Zbl 0851.11055
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