"Representation functions with prescribed rates of growth"
Submitted (2024+)
<arXiv>
Fix \(h\geq 2\), and let \(b_1\) , \(\ldots\), \(b_h\) be not necessarily distinct positive integers with \(\gcd = 1\). For any subset \(A\subseteq \mathbb{N}\), write \(r_{A}(n)\) (\(n\geq 0\)) for the number of solutions \((k_1,\ldots,k_h)\in A^h\) to
\[ b_1 k_1 + \cdots + b_h k_h = n. \]
For which functions \(F\) with \(F(n) \leq r_{\mathbb{N}}(n)\) can we find \(A\subseteq \mathbb{N}\) such that \(r_{A}(n)\sim F(n)\)? We show that if \(F\) is regularly varying, then there exists such an \(A\) provided \(\lim_{n\to\infty} F(n)/\log n = \infty\). If one only requires \(r_{A}(n) \asymp F(n)\), we show there exists such an \(A\) for every increasing \(F\) satisfying \(F(2x) \ll F(x)\) and \(\liminf_{n\to\infty} F(n)/\log n > 0\).
Lastly, we give a probabilistic heuristic for the following statement: if \(A\subseteq \mathbb{N}\) is such that \(\limsup_{n\to\infty} r_{A}(n)/\log n < 1\), then \(r_{A}(n) = 0\) for infinitely many \(n\).
"Zeros near \(s=1\) and the constant term of \(L'/L\) for \(L\)-functions in the Selberg class"
Submitted (2023+)
<arXiv>
Let \(\mathcal{L}(s) = \sum_{n=1}^{\infty} a_n n^{-s}\) be an \(L\)-function in the Selberg class, and \(q_{\mathcal{L}}\) its conductor. Let \(\ell_0(\mathcal{L})\) be the constant term of the Laurent expansion of \(\mathcal{L}'/\mathcal{L}\) at \(s=1\). We show that for certain families \(\mathcal{F}\) of \(L\)-functions in the Selberg class with polynomial Euler product:
If \(\mathcal{L}\in\mathcal{F}\) has no zeros \(\beta + i\gamma\) with \(\beta > 1 - \delta(\log q_{\mathcal{L}})^{-1}\), \(|\gamma| < (\log q_{\mathcal{L}})^{-1/2}\) for some absolute \(\delta >0\), then \(\Re(\ell_0(\mathcal{L})) \ll_{\mathcal{F}} \log q_{\mathcal{L}}\);
If \(\Re(\ell_0(\mathcal{L})) \ll \log q_{\mathcal{L}}\) for all \(\mathcal{L}\in \mathcal{F}\), then there is some absolute \(\delta > 0\) such that \(\mathcal{L}\) has no zeros \(\beta + i\gamma\) with \(\beta > 1 - \delta(\log q_{\mathcal{L}})^{-1}\), \(|\gamma| < (1-\beta)^{1/2}(\log q_{\mathcal{L}})^{-1/2}\).
This generalizes, for instance, the case of families of Dedekind zeta functions of number fields with bounded degree.
"On the size and structure of \(t\)-representable sumsets"
Submitted (2023+)
<arXiv>
Let \(A\subseteq \mathbb{Z}_{\geq 0}\) be a finite set with minimum element \(0\), maximum element \(m\), and \(\ell\) elements strictly 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 \(A\). We prove that \((hA)^{(t)}\) is "structured" for
\[ h \geq (1+o(1)) \frac{1}{e} m\ell t^{1/\ell} \]
(as \(\ell \to \infty\), \(t^{1/\ell} \to \infty\)), and prove a similar theorem on the structure of \(A\subseteq \mathbb{Z}^d\) for \(h\) sufficiently large. Moreover, we construct a family of sets \(A = A(m,\ell,t)\subseteq \mathbb{Z}_{\geq 0}\) for which \((hA)^{(t)}\) is not structured for \(h\ll m\ell t^{1/\ell}\).
"Knights are 24/13 times faster than the king"
Accepted (2024). To appear on Fibonacci Quarterly.
<arXiv>
On an infinite chess board, how much faster can the knight reach a square when compared to the king, in average? More generally, for coprime \(b>a \in \mathbb{Z}_{\geq 1}\) such that \(a+b\) is odd, define the \((a,b)\)-knight and the king as
\[ \mathrm{N}_{a,b}= \{(a,b), (b,a), (-a,b), (-b,a), (-b,-a), (-a,-b), (a,-b), (b, -a)\}, \]
\[ \mathrm{K}=\{(1,0), (1,1), (0,1), (-1,1), (-1,0), (-1,-1), (0,-1), (1,-1)\} \subseteq \mathbb{Z}^2, \]
respectively. One way to formulate this question is by asking for the average ratio, for \(\mathbf{p}\in \mathbb{Z}^2\) in a box, between \(\min\{h\in \mathbb{Z}_{\geq 1} ~|~ \mathbf{p}\in h\mathrm{N}\}\) and \(\min\{h\in \mathbb{Z}_{\geq 1} ~|~ \mathbf{p}\in h\mathrm{K}\}\), where \(hA = \{\mathbf{a}_1+\cdots+\mathbf{a}_h ~|~ \mathbf{a}_1,\ldots, \mathbf{a}_h \in A\}\) is the \(h\)-fold sumset of \(A\). We show that this ratio equals \(2(a+b)b^2/(a^2+3b^2)\).
"Classification of the conjugacy classes of \(\widetilde{\mathrm{SL}}(2,\mathbb{R})\)"
Accepted (2024). To appear on Expositiones Mathematicae.
<arXiv>
In this note, we classify the conjugacy classes of \(\widetilde{\mathrm{SL}}_2(\mathbb{R})\), the universal covering group of \(\mathrm{PSL}_2(\mathbb{R})\). For any non-central element \(\alpha \in \widetilde{\mathrm{SL}}_2(\mathbb{R})\), we show that its conjugacy class may be determined by three invariants:
Trace: the trace (valued in the set of positive real numbers \(\mathbb{R}_{+}\)) of its image \(\overline{\alpha}\) in \(\mathrm{PSL}_2(\mathbb{R})\);
Direction type: the sign behavior of the induced self-homeomorphism of \(\mathbb{R}\) determined by the lifting \(\widetilde{\mathrm{SL}}_2(\mathbb{R}) \curvearrowright \mathbb{R}\) of the action \(\mathrm{PSL}_2(\mathbb{R}) \curvearrowright \mathbb{S}^{1}\);
The function \(\ell^{\sharp}\): a conjugacy invariant length function introduced by S. Mochizuki [Res. Math. Sci. 3 (2016), 3:6].
"On Landau–Siegel zeros and heights of singular moduli"
Acta Arithmetica 201(1), 1–28 (2021)
<arXiv | DOI> [figures (python/bash scripts)]
Let \(\chi_D\) be the Dirichlet character associated to \(\mathbb{Q}(\sqrt{D})\) where \(D < 0\) is a fundamental discriminant. Improving Granville–Stark [DOI:10.1007/s002229900036], we show that
\[ \frac{L'}{L}(1,\chi_D) = \frac{1}{6}\, \mathrm{height}(j(\tau_D)) - \frac{1}{2}\log|D| + C + o_{D\to -\infty}(1), \]
where \(\tau_D = \frac 12(-\delta+\sqrt{D})\) for \(D \equiv \delta ~(\mathrm{mod}~4)\) and \(j(\cdot)\) is the \(j\)-invariant function with \(C = -1.057770\ldots\). Assuming the ``uniform'' \(abc\)-conjecture for number fields, we deduce that \(L(\beta,\chi_D)\ne 0\) with \(\beta ≥ 1 - \frac{\sqrt{5}\varphi + o(1)}{\log|D|}\) where \(\varphi = \frac{1+\sqrt{5}}{2}\), which we improve for smooth \(D\).
"An elementary heuristic for Hardy–Littlewood extended Goldbach's conjecture"
São Paulo Journal of Mathematical Sciences 14(1), 391–405 (2020)
<arXiv | DOI> [figures (python/bash scripts)]
The goal of this paper is to describe an elementary combinatorial heuristic that predicts Hardy & Littlewood's extended Goldbach's conjecture. We examine common features of other heuristics in additive prime number theory, such as Cramér's model and density-type arguments, both of which our heuristic draws from. Apart from the prime number theorem, our argument is entirely elementary, in the sense of not involving complex analysis. The idea is to model sums of two primes by a hypergeometric probability distribution, and then draw heuristic conclusions from its concentration behavior, which follows from Hoeffding-type bounds.
"An extension of the Erdős–Tetali theorem"
Random Structures & Algorithms 55(1), 173–214 (2019)
<arXiv | DOI>
Given a sequence \(\mathscr{A}=\{a_0 < a_1 < a_2\ldots\}\subseteq \mathbb{N}\), let \(r_{\mathscr{A},h}(n)\) denote the number of ways \(n\) can be written as the sum of \(h\) elements of \(\mathscr{A}\). Fixing \(h\geq 2\), we show that if \(f\) is a suitable real function (namely: locally integrable, \(O\)-regularly varying and of positive increase) satisfying
\[ x^{1/h}\log(x)^{1/h} \ll f(x) \ll \frac{x^{1/(h-1)}}{\log(x)^{\varepsilon}} \text{ for some } \varepsilon > 0, \]
then there must exist \(\mathscr{A}\subseteq\mathbb{N}\) with \(|\mathscr{A}\cap [0,x]|=\Theta(f(x))\) for which \(r_{\mathscr{A},h+\ell}(n) = \Theta(f(n)^{h+\ell}/n)\) for all \(\ell \geq 0\). Furthermore, for \(h=2\) this condition can be weakened to \(x^{1/2}\log(x)^{1/2} \ll f(x) \ll x\).
The proof is somewhat technical and the methods rely on ideas from regular variation theory, which are presented in an appendix with a view towards the general theory of additive bases. We also mention an application of these ideas to Schnirelmann's method.