Sums of four squareful numbers

Shute, Alec

doi:10.48550/arxiv.2104.06966

Cited by 2 publications

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“…Van Valckenborgh proves in [19,Theorem 1.1] that for any n 3, we have N n (B) ∼ cB n/2 as B → ∞, for an explicit constant c > 0. In [16], we extend the treatment to handle the case n = 2. Work of Browning and Yamagishi in [3] considers a more general orbifold (P n , D), where the D i are as above, and D = n+1 i=0 (1 − 1 m i )D i for integers m 0 , .…”

Section: Introductionmentioning

confidence: 99%

“…With this approach, the leading constant for N n (B) is expressed as an infinite sum of constants c y arising from Manin's conjecture applied to N + y (B). This is the point of view taken in [19,Section 5] for n 3, and it is also how we express the leading constant in [16] for the case n = 2. When n = 1, it leads to the following prediction in [2, Conjecture 1.1].…”

Section: Introductionmentioning

confidence: 99%

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On the leading constant in the Manin-type conjecture for Campana points

Shute¹

2021

Preprint

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We compare the Manin-type conjecture for Campana points recently formulated by Pieropan, Smeets, Tanimoto and Várilly-Alvarado with an alternative prediction of Browning and Van Valckenborgh in the special case of the orbifold (P 1 , D), where. We find that the two predicted leading constants do not agree, and we discuss whether thin sets could explain this discrepancy. Motivated by this, we provide a counterexample to the Manin-type conjecture for Campana points, by considering orbifolds corresponding to squareful values of binary quadratic forms.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On the leading constant in the Manin-type conjecture for Campana points

Shute¹

2021

Preprint

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show abstract

“…With this approach, the leading constant for N n (B) is expressed as an infinite sum of constants c y arising from Manin's conjecture applied to N + y (B). This is the point of view taken in [22, Section 5] for n ≥ 3, and it is also how we express the leading constant in [19] for the case n = 2. When n = 1, it leads to the following prediction [2, Conjecture 1.1].…”

mentioning

confidence: 99%

“…Van Valckenborgh [22,Theorem 1.1] proves that for any n ≥ 3, we have N n (B) ∼ cB n/2 as B → ∞, for an explicit constant c > 0. In [19], we extend the treatment to handle the case n = 2. Work of Browning and Yamagishi [3] considers a more general orbifold (P n , D), where the D i are as above, and D = n+1 i=0 (1 − 1/m i )D i for integers m 0 , .…”

mentioning

confidence: 99%

On the leading constant in the Manin-type conjecture for Campana points

Shute

2022

Acta Arith.

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On the leading constant in the Manin-type conjecture for Campana points by Alec Shute (Klosterneuburg)1. Introduction. The study of Campana points is an emerging area of interest in arithmetic geometry as a way to interpolate between rational and integral points. Campana orbifolds, first introduced in [4] and [5], consist of a variety X and a weighted boundary divisor D of X. The Campana points associated to the orbifold (X, D) can be viewed as rational points of X that are integral with respect to D. In the recent paper [16], Pieropan, Smeets, Tanimoto and Várilly-Alvarado formulate a Manin-type conjecture for the quantitative study of Campana points on Fano Campana orbifolds, which henceforth we shall refer to as the PSTV-A conjecture. The authors establish their conjecture in the special case of vector group compactifications, using the height zeta function method developed by Chambert-Loir and Tschinkel [7], [8].The arithmetic study of Campana points is still in its early stages. Initial results in [2], [22] and [3], which predate the formulation of the PSTV-A conjecture, concern squareful and m-full values of hyperplanes of P n+1 . (We recall that a non-zero integer z is m-full if for any prime p dividing z, we have p m | z, and squareful if it is 2-full.) Following discussions in the Spring 2006 MSRI program on rational and integral points on higherdimensional varieties, Poonen [17] posed the problem of finding the number of coprime integers z 0 , z 1 such that z 0 , z 1 and z 0 + z 1 are all squareful and bounded by B. In the language of the PSTV-A conjecture, this corresponds to counting Campana points on the orbifold (P 1 , D), where D is the divisorUpper and lower bounds for this problem were obtained by Browning and Van Valckenborgh [2], but finding an asymptotic formula remains wide open. Van Valckenborgh [22] considers a higher-dimensional

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Sums of four squareful numbers

Cited by 2 publications

References 22 publications

On the leading constant in the Manin-type conjecture for Campana points

On the leading constant in the Manin-type conjecture for Campana points

On the leading constant in the Manin-type conjecture for Campana points

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