An effective bound for generalised Diophantine \(m\)-tuples. (English) Zbl 07828706
For any integer \(k\geq 2\), we say that \(S\) is a generalised Diophantine \(m\)-tuple with property \(D_k(n)\) if \(a_ia_j+n\) is a perfect \(k\)-th power for all \(i,j\) with \(1\leq i<j\leq m\), and set \(M_k(n):=\sup \{ |S|:\, S\) satisfies property \(D_k(n)\}\). For \(k=2\) we write \(D(n)\), \(M(n)\) instead of \(D_k(n)\), \(M_k(n)\) and omit the adjective ‘generalized’. Further, for \(L>0\) we define \(M_k(n,L):=\sup \{ |S\cap [ |n|^L,\infty )|:\, S\) satisfies property \(D_k(n)\}\).
There is a lot of work with upper bounds for \(M_k(n)\). For instance, A. Dujella [J. Reine Angew. Math. 566, 183–214 (2004; Zbl 1037.11019)] obtained \(M(n)\ll\log |n|\), R. Becker and M. R. Murty [Glas. Mat., III. Ser. 54, No. 1, 65–75 (2019; Zbl 1455.11048)] derived the more precise bound \(M(n)\leq 2.6071\log |n|+O\big(\log |n|/(\log\log |n|)^2\big)\), A. Bérczes et al. [Publ. Math. Debr. 79, No. 3–4, 325–339 (2011; Zbl 1249.11036)] proved \(M_k(n)\leq 2|n|^5+3\) for \(k\geq 5\), and lastly, for large positive \(n\) and any fixed \(k\), A. B. Dixit et al. [Proc. Am. Math. Soc. 150, No. 4, 1455–1465 (2022; Zbl 1489.11050)] improved this for \(k\geq 3\) and \(n\to\infty\) to \(M_k(n,L)\ll_{k,L} 1\) for \(L\geq 3\) and \(M_k(n) \ll _k \log n\).
In the paper under review, the authors extend this to \(n<0\) and prove the following more precise result for \(k\) not growing to quickly in terms of \(n\):
Theorem. Let \(k\geq 3\) be a positive integer. Then
(a) For \(L\geq 3\), \(M_k(n,L)\leq 2^{28}\log (2k)\log (2\log (2k))+14\).
(b) Suppose \(n\) and \(k\) vary such that as \(|n|\to\infty\) and \(k=o(\log\log |n|)\), then \(M_k(n)\leq 3\phi (k)\log |n|+O(\big( (\phi (k)^2\log |n|/\log\log |n|\big)\), where \(\phi (n)\) denotes Euler’s totient function.
To prove this result, the authors extend a gap principle due to K. Gyarmati [Acta Arith. 97, No. 1, 53–65 (2001; Zbl 0986.11016)], and prove (a) using a quantitative version of Roth’s theorem on the approximation of algebraic numbers by rationals, due to the reviewer [J. Math. Sci., New York 171, No. 6, 824–837 (2010; Zbl 1282.11085)] and subsequently (b) using P. X. Gallagher’s larger sieve [Acta Arith. 18, 77–81 (1971; Zbl 0231.10028)].
There is a lot of work with upper bounds for \(M_k(n)\). For instance, A. Dujella [J. Reine Angew. Math. 566, 183–214 (2004; Zbl 1037.11019)] obtained \(M(n)\ll\log |n|\), R. Becker and M. R. Murty [Glas. Mat., III. Ser. 54, No. 1, 65–75 (2019; Zbl 1455.11048)] derived the more precise bound \(M(n)\leq 2.6071\log |n|+O\big(\log |n|/(\log\log |n|)^2\big)\), A. Bérczes et al. [Publ. Math. Debr. 79, No. 3–4, 325–339 (2011; Zbl 1249.11036)] proved \(M_k(n)\leq 2|n|^5+3\) for \(k\geq 5\), and lastly, for large positive \(n\) and any fixed \(k\), A. B. Dixit et al. [Proc. Am. Math. Soc. 150, No. 4, 1455–1465 (2022; Zbl 1489.11050)] improved this for \(k\geq 3\) and \(n\to\infty\) to \(M_k(n,L)\ll_{k,L} 1\) for \(L\geq 3\) and \(M_k(n) \ll _k \log n\).
In the paper under review, the authors extend this to \(n<0\) and prove the following more precise result for \(k\) not growing to quickly in terms of \(n\):
Theorem. Let \(k\geq 3\) be a positive integer. Then
(a) For \(L\geq 3\), \(M_k(n,L)\leq 2^{28}\log (2k)\log (2\log (2k))+14\).
(b) Suppose \(n\) and \(k\) vary such that as \(|n|\to\infty\) and \(k=o(\log\log |n|)\), then \(M_k(n)\leq 3\phi (k)\log |n|+O(\big( (\phi (k)^2\log |n|/\log\log |n|\big)\), where \(\phi (n)\) denotes Euler’s totient function.
To prove this result, the authors extend a gap principle due to K. Gyarmati [Acta Arith. 97, No. 1, 53–65 (2001; Zbl 0986.11016)], and prove (a) using a quantitative version of Roth’s theorem on the approximation of algebraic numbers by rationals, due to the reviewer [J. Math. Sci., New York 171, No. 6, 824–837 (2010; Zbl 1282.11085)] and subsequently (b) using P. X. Gallagher’s larger sieve [Acta Arith. 18, 77–81 (1971; Zbl 0231.10028)].
Reviewer: Jan-Hendrik Evertse (Leiden)
MSC:
| 11D45 | Counting solutions of Diophantine equations |
| 11D72 | Diophantine equations in many variables |
| 11N36 | Applications of sieve methods |
Citations:
Zbl 1489.11050; Zbl 1037.11019; Zbl 1455.11048; Zbl 1249.11036; Zbl 0986.11016; Zbl 1282.11085; Zbl 0231.10028References:
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