Heath-Brown (2009): sums and differences of three kth powers

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theorem_1: For a non-singular integral ternary form of degree at least 3 and natural N <<_F B^(3/13), the integer points with |F(x)| <= N and B/2 < max|x_i| <= B lie on O_F(B^(9/10) N^(1/10)) conics, and the count outside linear families is O(B^(9/10+eps) N^(1/10)).

theorem_2: For a fixed non-singular integral ternary form of degree k >= 3 and natural N <<_F B, the solutions of F(x) = N with B/2 < max|x_i| <= B outside polynomial families of degree at most [k/10] number O_F(B^(10/k)).

theorem_3: For fixed nonzero integer h and k >= 3, the number of primes p <= X for which p^k + h is (k-1)-free is c_{h,k} Li(X) + o(X/log X), with an explicit Euler product c_{h,k}.

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D. R. Heath-Brown, Sums and differences of three kth powers, Journal of Number Theory 129 (2009), no. 6, 1579-1594. DOI: 10.1016/j.jnt.2009.01.012. The journal first page records receipt on 26 June 2008, revision on 22 January 2009, and online publication on 20 March 2009.

Source versions

The edition read is the journal version of record, 16 PDF pages, PDF page 1 being printed page 1579; it prints "© 2009 Elsevier Inc. All rights reserved." on p. 1579. The earlier arXiv:0806.4330v1 (26 June 2008, 17 pages) was also read for the introduction, under arXiv's non-exclusive distribution license.

The versions differ in the counting region of $\mathcal N(B;N,F,d)$: the journal (p. 1580) counts $B/2<\max_i|x_i|\leq B$, v1 (p. 1) counts $\max_i|x_i|\leq B$. Theorems 1 and 2 display the same bounds in both. Neither version says in its definition that a parametric family must be nonconstant; the proof counts only nonconstant families, as the Theorem 2 page records.

Results

For a non-singular form $F\in\mathbb Z[x_1,x_2,x_3]$ of degree $k\geq3$, $\mathcal N(B;N,F,d)$ counts integer solutions of $F(\mathbf x)=N$ in the shell $B/2<\max_i|x_i|\leq B$ off polynomial parameterizations of degree at most $d$ (p. 1580).

• Theorem 1 (p. 1580): for $N\ll_F B^{3/13}$ the integer points with $|F(\mathbf x)|\leq N$ in the shell lie on $O_F(B^{9/10}N^{1/10})$ conics, and $\mathcal N(B;N,F,1)=O_{F,\varepsilon}(B^{9/10+\varepsilon}N^{1/10})$.
• Theorem 2 (p. 1580): for $N\ll_F B$, $\mathcal N(B;N,F,\lfloor k/10\rfloor)\ll_F B^{10/k}$, and the number of essentially different families of degree at most $\lfloor k/10\rfloor$ is bounded in terms of $k$ alone.
• Theorem 3 (pp. 1581--1582): for fixed $h\neq0$ and $k\geq3$, the primes $p\leq X$ with $p^k+h$ free of $(k-1)$th powers number $c_{h,k}\mathrm{Li}(X)+o(X(\log X)^{-1})$.

The proofs use the real-variable determinant method: Section 2 (pp. 1582--1585) covers the points by curves of low degree, Section 3 (pp. 1585--1588) proves Lemmas 1 to 3, Section 4 (pp. 1588--1592) counts points on those curves for Theorems 1 and 2, and Section 5 (pp. 1592--1593) proves Theorem 3.

Read status: claims checked for Theorems 1 to 3, read clause by clause on the page images of the journal print and, for Theorems 1 and 2, of arXiv v1; the proofs were read for structure only. Nothing here is independently reviewed.

Bears on. Problem 477: the paper does not mention the problem; the pipeline-math manuscript restates Theorem 2 as a whole-box bound and cites it in the proof of its Proposition 1.6, a step in its construction of a set $A$ such that every integer is uniquely $a+m^{13}$ with $a\in A$, $m\in\mathbb Z$. The paper itself proves nothing about additive complements.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.