Christian Táfula

Invited talks

"From ABC to L: On singular moduli and Siegel zeros"
1. [2020/08] 2nd Kyoto–Hefei Workshop on Arithmetic Geometry <webpage>
2. [2021/07] MOBIUS ANT Seminar <webpage>
3. [2021/09] Inter-universal Teichmüller Theory (IUT) Summit <handout | webpage>
4. [2021/12] Seminario Latinoamericano de Teoría de Números <handout | webpage> (pt-BR)
  
  In 2000, using analytical, algebraic, and arithmetical ideas, Granville and Stark showed that the "uniform" ABC for number fields implies that odd Dirichlet L-functions have no "Siegel zeroes", which are a severe type of (not yet unconditionally ruled out) counterexample to the Generalized Riemann Hypothesis. In this talk we are going to focus on the structure of their main argument, and discuss recent work that allows us to get more precise relations between the analysis (zero-free regions of L-functions) and the arithmetics (heights of singular moduli).
"On the structure of \(t\)-representable sumsets"
1. [2023/04] MOBIUS ANT Seminar <webpage>
2. [2023/05] Combinatorial and Additive Number Theory (CANT 2023) <handout | webpage>
  
  Let \(A\subseteq \mathbb{Z}_{\geq 0}\) be a finite set with minimal element \(0\), maximum element \(m\), and \(\ell\) elements in between. Write \((hA)^{(t)}\) for the set of integers that can be written in at least \(t\) ways as a sum of \(h\) elements of \(A\). In 1970, Nathanson showed that \(hA = (hA)^{(1)}\) enjoys a notion of "structure" for large \(h\), allowing us to determine \(hA\) in a relatively simple fashion. This was revisited by Granville, Shakan and Walker in 2020, who showed that \(hA\) is structured for every \(h \geq m - \ell\), and that finite sets in \(\mathbb{Z}^d\) also have a similar notion of structure.
  
  Also in 2020 (50 years later!), Nathanson showed that \((hA)^{(t)}\) enjoys a notion of structure as well for large \(h\). In this talk, we will study the problem of giving upper bounds to how large h should be, showing that \((hA)^{(t)}\) is structured if \(h \gtrsim \tfrac{1}{e} m\ell t^{1/\ell}\). This estimate is asymptotically sharp, as we can construct a family of sets \(A = A(m, \ell, t) \subseteq \mathbb{Z}_{\geq 0}\) for which \((hA)^{(t)}\) is not structured for \(h \leq (1 - o(1)) \tfrac{1}{e} m\ell t^{1/\ell}\).
  
  By the end, we briefly discuss how \((hA)^{(t)}\) for finite sets \(A \subseteq \mathbb{Z}^{d}\) enjoys a similar notion of structure for large \(h\).
"Representation functions with prescribed rates of growth"
1. [2024/04] MOBIUS ANT Seminar <webpage>
  
  Let \(h\geq 2\), and \(b_1\), \(\ldots\), \(b_h\) be positive integers with \(\gcd = 1\). For a set \(A\subseteq \mathbb{N}\), denote by \(r_A(n)\) the number of solutions to the equation \[ b_1 k_1 + ... + b_h k_h = n \] with \(k_1\), \(\ldots\), \(k_h\in A\). For which functions \(F\) can we find \(A\) such that \(r_A(n) \sim F(n)\)? Or \(r_A(n)\asymp F(n)\)? In the asymptotic case, we show that for every \(F\) "regularly varying" in the range \[ \omega(\log x) = F(x) \leq (\tfrac{1}{(h-1)!b_1...b_h} + o(1)) x^{h-1} \] (where \(f=\omega(g)\) if \(g=o(f)\)), there is \(A\) such that \(r_A(n) \sim F(n)\). In the order of magnitude case, there is \(A\) with \(r_A(n)\asymp F(x)\) for every \(F\) non-decreasing such that \(F(2x)\ll F(x)\) in the range \(\log x \ll F(x) \ll x^{h-1}\). This extends earlier work of Erdős–Tetali and Vu, and addresses a question raised by Nathanson on which functions can be the \(r_A\) of some \(A\). If time allows, we will show for \(1\ll F(x) = o(\log x)\) a probabilistic heuristic for why there should not exist \(A\) with \(r_A(n) \asymp F(n)\), agreeing with a conjecture of Erdős.