@article{11636,
  abstract     = {In [3], Poonen and Slavov recently developed a novel approach to Bertini irreducibility theorems over an arbitrary field, based on random hyperplane slicing. In this paper, we extend their work by proving an analogous bound for the dimension of the exceptional locus in the setting of linear subspaces of higher codimensions.},
  author       = {Kmentt, Philip and Shute, Alec L},
  issn         = {1090-2465},
  journal      = {Finite Fields and their Applications},
  number       = {10},
  publisher    = {Elsevier},
  title        = {{The Bertini irreducibility theorem for higher codimensional slices}},
  doi          = {10.1016/j.ffa.2022.102085},
  volume       = {83},
  year         = {2022},
}

@article{17058,
  abstract     = {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 (P1,D), where D=1/2[0]+1/2[1]+1/2[∞]. 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.},
  author       = {Shute, Alec L},
  issn         = {1730-6264},
  journal      = {Acta Arithmetica},
  number       = {4},
  pages        = {317--346},
  publisher    = {Institute of Mathematics},
  title        = {{On the leading constant in the Manin-type conjecture for Campana points}},
  doi          = {10.4064/aa210430-1-7},
  volume       = {204},
  year         = {2022},
}

@phdthesis{12072,
  abstract     = {In this thesis, we study two of the most important questions in Arithmetic geometry: that of the existence and density of solutions to Diophantine equations. In order for a Diophantine equation to have any solutions over the rational numbers, it must have solutions everywhere locally, i.e., over R and over Qp for every prime p. The converse, called the Hasse principle, is known to fail in general. However, it is still a central question in Arithmetic geometry to determine for which varieties the Hasse principle does hold. In this work, we establish the Hasse principle for a wide new family of varieties of the form f(t) = NK/Q(x) ̸= 0, where f is a polynomial with integer coefficients and NK/Q denotes the norm
form associated to a number field K. Our results cover products of arbitrarily many linear, quadratic or cubic factors, and generalise an argument of Irving [69], which makes use of the beta sieve of Rosser and Iwaniec. We also demonstrate how our main sieve results can be applied to treat new cases of a conjecture of Harpaz and Wittenberg on locally split values of polynomials over number fields, and discuss consequences for rational points in fibrations.
In the second question, about the density of solutions, one defines a height function and seeks to estimate asymptotically the number of points of height bounded by B as B → ∞. Traditionally, one either counts rational points, or
integral points with respect to a suitable model. However, in this thesis, we study an emerging area of interest in Arithmetic geometry known as Campana points, which in some sense interpolate between rational and integral points.
More precisely, we count the number of nonzero integers z1, z2, z3 such that gcd(z1, z2, z3) = 1, and z1, z2, z3, z1 + z2 + z3 are all squareful and bounded by B. Using the circle method, we obtain an asymptotic formula which agrees in
the power of B and log B with a bold new generalisation of Manin’s conjecture to the setting of Campana points, recently formulated by Pieropan, Smeets, Tanimoto and Várilly-Alvarado [96]. However, in this thesis we also provide the first known counterexamples to leading constant predicted by their conjecture. },
  author       = {Shute, Alec L},
  isbn         = {978-3-99078-023-7},
  issn         = {2663-337X},
  pages        = {208},
  publisher    = {Institute of Science and Technology Austria},
  title        = {{Existence and density problems in Diophantine geometry: From norm forms to Campana points}},
  doi          = {10.15479/at:ista:12072},
  year         = {2022},
}

@unpublished{12077,
  abstract     = {We compare the Manin-type conjecture for Campana points recently formulated
by Pieropan, Smeets, Tanimoto and V\'{a}rilly-Alvarado with an alternative
prediction of Browning and Van Valckenborgh in the special case of the orbifold
$(\mathbb{P}^1,D)$, where $D =\frac{1}{2}[0]+\frac{1}{2}[1]+\frac{1}{2}[\infty]$. 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.},
  author       = {Shute, Alec L},
  booktitle    = {arXiv},
  title        = {{On the leading constant in the Manin-type conjecture for Campana points}},
  doi          = {10.48550/arXiv.2104.14946},
  year         = {2021},
}

@unpublished{12076,
  abstract     = {We find an asymptotic formula for the number of primitive vectors $(z_1,\ldots,z_4)\in (\mathbb{Z}_{\neq 0})^4$ such that $z_1,\ldots, z_4$ are all squareful and bounded by $B$, and $z_1+\cdots + z_4 = 0$. Our result agrees in the power of $B$ and $\log B$ with the Campana-Manin conjecture of Pieropan, Smeets, Tanimoto and V\'{a}rilly-Alvarado.},
  author       = {Shute, Alec L},
  booktitle    = {arXiv},
  title        = {{Sums of four squareful numbers}},
  doi          = {10.48550/arXiv.2104.06966},
  year         = {2021},
}