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- Brandes, Julia, et al.
(author)
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On generating functions in additive number theory, II: lower-order terms and applications to PDEs
- 2021
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In: Mathematische Annalen. - : Springer Science and Business Media LLC. - 0025-5831 .- 1432-1807. ; 379, s. 347-76
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Journal article (peer-reviewed)abstract
- We obtain asymptotics for sums of the form Sigma(p)(n=1) e(alpha(k) n(k) + alpha(1)n), involving lower order main terms. As an application, we show that for almost all alpha(2) is an element of [0, 1) one has sup(alpha 1 is an element of[0,1)) | Sigma(1 <= n <= P) e(alpha(1)(n(3) + n) + alpha(2)n(3))| << P3/4+epsilon, and that in a suitable sense this is best possible. This allows us to improve bounds for the fractal dimension of solutions to the Schrodinger and Airy equations.
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| 2. |
- Brandes, Julia, 1986, et al.
(author)
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Simultaneous Additive Equations: Repeated and Differing Degrees
- 2017
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In: Canadian Journal of Mathematics. - 1496-4279 .- 0008-414X. ; 69:2, s. 258-283
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Journal article (peer-reviewed)abstract
- We obtain bounds for the number of variables required to establish Hasse principles, both for the existence of solutions and for asymptotic formula, for systems of additive equations containing forms of differing degree but also multiple forms of like degree. Apart from the very general estimates of Schmidt and Browning-Heath-Brown, which give weak results when specialized to the diagonal situation, this is the first result on such "hybrid" systems. We also obtain specialized results for systems of quadratic and cubic forms, where we are able to take advantage of some of the stronger methods available in that setting. In particular, we achieve essentially square root cancellation for systems consisting of one cubic and r quadratic equations.
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