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pro vyhledávání: '"Brown, D R"'
Autor:
Heath-Brown, D. R.
We improve the standard Weyl estimate for quartic exponential sums in which the argument is a quadratic irrational. Specifically we show that \[\sum_{n\le N} e(\alpha n^4)\ll_{\ep,\alpha}N^{5/6+\ep}\] for any $\ep>0$ and any quadratic irrational $\al
Externí odkaz:
http://arxiv.org/abs/2312.14531
Autor:
Heath-Brown, D. R.
We present a general method for handling problems that ask for the equidistribution of solutions to equations involving $m^2+n^2$, and illustrate it by considering $p+m^2+n^2=N$.
Comment: Revised version incorporating several corrections
Comment: Revised version incorporating several corrections
Externí odkaz:
http://arxiv.org/abs/2212.12587
Autor:
Heath-Brown, D. R.
We examine the counting function for rational points on conics, and show how the point where the asymptotic behaviour begins depends on the size of the smallest zero.
Externí odkaz:
http://arxiv.org/abs/2203.13889
Autor:
Heath-Brown, D. R.
We show that \[\sum_{\substack{p_n\le x\\ p_{n+1}-p_n\ge\sqrt{p_n}}}(p_{n+1}-p_n)\ll_{\varepsilon} x^{3/5+\varepsilon}\] for any fixed $\varepsilon>0$. This improves a result of Matom\"{a}ki, in which the exponent was $2/3$.
Externí odkaz:
http://arxiv.org/abs/1906.09555
Akademický článek
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Autor:
Heath-Brown, D R
We show that large gaps between smooth numbers are infrequent. The key new tool is a novel mean value bound for a special type of Dirichlet polynomial.
Comment: To appear in Acta Arithmetica
Comment: To appear in Acta Arithmetica
Externí odkaz:
http://arxiv.org/abs/1808.02947
Autor:
Browning, T. D., Heath-Brown, D. R.
Publikováno v:
Duke Math. J. 169, no. 16 (2020), 3099-3165
An asymptotic formula is established for the number of rational points of bounded anticanonical height which lie on a certain Zariski dense subset of the biprojective hypersurface $x_1y_1^2+\dots+x_4y_4^2=0$ in $\mathbb{P}^3\times\mathbb{P}^3$. This
Externí odkaz:
http://arxiv.org/abs/1805.10715
Autor:
Browning, T. D., Heath-Brown, D. R.
Publikováno v:
Discrete Analysis 2018:15
We give an upper bound for the number of rational points of height at most $B$, lying on a surface defined by a quadratic form $Q$. The bound shows an explicit dependence on $Q$. It is optimal with respect to $B$, and is also optimal for typical form
Externí odkaz:
http://arxiv.org/abs/1801.00979
Autor:
Heath-Brown, D. R., Micheli, Giacomo
For a set $S$ of quadratic polynomials over a finite field, let $C$ be the (infinite) set of arbitrary compositions of elements in $S$. In this paper we show that there are examples with arbitrarily large $S$ such that every polynomial in $C$ is irre
Externí odkaz:
http://arxiv.org/abs/1701.05031
Autor:
Heath-Brown, D. R.
For a finite field of odd cardinality $q$, we show that the sequence of iterates of $aX^2+c$, starting at $0$, always recurs after $O(q/\log\log q)$ steps. For $X^2+1$ the same is true for any starting value. We suggest that the traditional "Birthday
Externí odkaz:
http://arxiv.org/abs/1701.02707