arXiv subjects

Christian Táfula

Publications and source records attributed to Christian Táfula.

17 recordsLinked to original sources

arXiv2026

Quantitative linear independence for square roots

We consider the problem of finding lower bounds for integer linear combinations of $\sqrt{a_1},\ldots,\sqrt{a_K}$, where $a_1,\ldots,a_K$ are positive integers such that their square roots are linearly independent over the rationals. We use a probabilistic approach and prove that for $K\geq 8$ and nonzero integers $m_1,\ldots,m_K$, $$ \bigg|\sum_{n\leq K} m_n\sqrt{a_n}\bigg| > e^{\frac{2^{K-1}-1}{K} - \frac{1}{2}} \bigg(\max_{n\leq K}|m_n|\sqrt{a_n}\cdot \sqrt{K}\bigg)^{-(2^{K-1}-1)}. $$ This inequality improves the dependence on $K$ in the classical product bound.

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arXiv2026

Bounded asymptotic bases for linear forms

For a vector of positive integers $\mathbf{b} = (b_1,\ldots,b_h)$ with $\gcd(b_1,\ldots,b_h) = 1$, we study sets $A \subseteq \mathbb{N}$ for which every sufficiently large integer has a bounded positive number of representations \[ n = b_1 x_1 + \cdots + b_h x_h \qquad (x_1,\ldots,x_h\in A). \] We prove that such a set exists for every binary vector $\mathbf{b} \neq (1,1)$, and for some general higher-dimensional families, including $\mathbf{b} = (u_1, p^d u_2, \ldots, p^{(h-1)d} u_h)$ where $p\nmid u_1\cdots u_h$.

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arXiv2026

An extension of the Erdős-Tetali theorem

Given a sequence $\mathscr{A}=\{a_0 0, \] then there must exist $\mathscr{A}\subseteq\mathbb{N}$ with $|\mathscr{A}\cap [0,x]|=Θ(f(x))$ for which $r_{\mathscr{A},h+\ell}(n) = Θ(f(n)^{h+\ell}/n)$ for all $\ell \geq 0$. Furthermore, for $h=2$ the same conclusion holds under $x^{1/2}\log(x)^{1/2} \ll f(x) \ll x$. 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. Corrections to the published version are highlighted in red.

math.CO↗

arXiv2026

The multiplication table problem in large dimensions

For $N\geq 2$ and $k\geq 1$, let $M_k(N):=\#\{x_1\cdots x_k : x_i\in\{1,\ldots,N\}\text{ for all } i\}$ be the $k$-dimensional multiplication table. Given $N$, Khovanskii's theorem implies that $M_k(N)$ agrees, for all sufficiently large $k$, with a polynomial in $k$ of degree $π(N)$. We determine the asymptotic size of its leading coefficient, proving that, as $N\to\infty$, with $k$ sufficiently large relative to $N$, \[ M_k(N) = \exp\bigg((2π+o(1))\frac{\sqrt{N}}{\log N}\bigg)\frac{k^{π(N)}}{π(N)!}. \] We also study the analogous problem when the factors are restricted to $y$-smooth integers. For $y=o(\log N)$, we prove that the number of distinct products of $k$ such integers up to $N$ is asymptotic to the number of $y$-smooth integers up to $N^k$, uniformly for $k\geq 1$.

math.NT↗

arXiv2026

Infinite Sidon-type sets for zero-sum linear forms

Let $h \geq 2$, and let $\mathbf{b} = (b_1,\dots,b_h)\in \mathbb{Z}^h$ be a zero-sum vector with nonzero coordinates. For a set $A=\{a_1<a_2<\cdots\}\subseteq\mathbb{N}$, let $r_{A,\mathbf{b}}(n)$ denote the number of $h$-tuples $(x_1,\ldots,x_h)$ of pairwise distinct elements of $A$ satisfying $b_1x_1+\cdots+b_hx_h=n$. We study density restrictions on sets $A$ for which these representation counts remain small, obtaining analogues of the classical density theorem for infinite Sidon sets. In the case $\mathbf{b} = (c_1,-c_1,\dots,c_k,-c_k)$, we prove that if $A(x)/(x/\log x)^{1/2k}\to\infty$, then $\frac{1}{x}\sum_{|n|\leq x} r_{A,\mathbf{b}}(n)\to\infty$, whereas if $A(x)\gg x^{1/2k}$, then $\frac{1}{x}\sum_{|n|\leq x} r_{A,\mathbf{b}}(n)\gg\log x$. This recovers Chen's theorem on $B_{2k}$-sequences. For general zero-sum vectors $\mathbf{b}$, we prove analogous bounds under gap conditions: if $a_{n+1}-a_n=o(n^{h-1}\log n)$, then $\frac{1}{x}\sum_{|n|\leq x} r_{A,\mathbf{b}}(n)\to\infty$, whereas if $a_{n+1}-a_n\ll n^{h-1}$, then $\frac{1}{x}\sum_{|n|\leq x} r_{A,\mathbf{b}}(n)\gg\log x$.

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arXiv2026

On Vu's restricted box estimate in Waring's problem

In 2000, Vu proved that the number of solutions of $x_1^k + \cdots + x_s^k = N$ in an arbitrary box satisfies the expected Hardy--Littlewood upper bound with a power-saving error term, for $s \geq O(8^k k^3)$. We show that one may take $s\geq k^2 - k + O(\sqrt{k})$.

math.NT↗

arXiv2026

Thin subbases of Piatetski-Shapiro sequences

For a non-integral real number $c>1$, let $\mathbb{N}_{(c)}:=\{\lfloor n^c\rfloor ~|~ n\in\mathbb{N}\}$. We show that $\mathbb{N}_{(c)}$ contains thin subbases of every order $h\geq 5$ when $1 2$. In fact, for every regularly varying function $F$ such that \[ \frac{F(x)}{\log x}\to\infty\quad\text{ and } \quad F(x)\leq (1+o(1))\frac{Γ(1+1/c)^h}{Γ(h/c)} x^{h/c-1}, \] there exists $A\subseteq\mathbb{N}_{(c)}$ with $r_{A,h}(n)\sim F(n)$. We also establish analogous results for $k$-th powers of Piatetski-Shapiro numbers and Piatetski-Shapiro primes for small $c$.

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arXiv2026

Waring and Waring-Goldbach subbases with prescribed representation function

Let $h\geq 2$. For $A\subseteq \mathbb{N}$ write \[ r_{A,h}(n) := \#\{(x_1,\ldots,x_h)\in A^h ~|~ x_1+\cdots+x_h=n\}. \] We prove a general probabilistic subbasis principle: assuming an asymptotic for a weighted $h$-fold representation sum over a basis $B$, there exist subbases $A\subseteq B$ whose representation function $r_{A,h}(n)$ has prescribed regularly varying growth. We apply this to $k$-th powers $\mathbb{N}^k$ and to $k$-th powers of primes $\mathbb{P}^k$. For $h \geq k^2-k+O(\sqrt{k})$, we show that every regularly varying function $F$ with $F(x)/\log x\to\infty$ in the admissible range is realized, with the expected singular series factor. In particular, there exists $A\subseteq \mathbb{N}^k$ such that \[ r_{A,h}(n)\sim \mathfrak{S}_{k,h}(n) F(n). \] Moreover, in the prime setting we obtain thin subbases $A\subseteq \mathbb{P}^k$ with $r_{A,h}(n)\asymp \log n$ for $n$ in the admissible congruence classes.

math.NT↗

arXiv2026

Representation functions with prescribed rates of growth

Fix an integer $h \geq 2$, and let $b_1, \ldots, b_h$ be (not necessarily distinct) positive integers with $\gcd(b_1, \ldots, b_h) = 1$. For any subset $A \subseteq \mathbb{N}$, let $r_A(n)$ denote the number of solutions $(k_1, \ldots, k_h) \in A^h$ to the equation \[ b_1 k_1 + \cdots + b_h k_h = n. \] Given a function $F$ satisfying $F(n) \leq r_{\mathbb{N}}(n)$, we ask: when does there exist a set $A \subseteq \mathbb{N}$ such that $r_A(n) \sim F(n)$? We prove that this is always possible when $F$ is regularly varying and satisfies $\lim_{n\to\infty} F(n)/\log n = \infty$. If one only requires $r_A(n) \asymp F(n)$, much weaker regularity conditions suffice: we show such a set $A$ exists for every increasing function $F$ satisfying $F(2x) \ll F(x)$ and $\log x \ll F(x) \ll x^{h-1}$. Finally, we give a probabilistic heuristic supporting the following: if $A \subseteq \mathbb{N}$ satisfies $\limsup_{n\to\infty} r_A(n)/\log n < 1$, then $r_A(n) = 0$ for infinitely many $n$.

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arXiv2025

A note on the Cramér-Granville model

We show the existence of a set $A\subseteq \mathbb{Z}_{\geq 2}$ satisfying the estimates of the Bateman--Horn conjecture, Goldbach's conjecture, and also \[ \#\{p\leq x \text{ prime} ~|~ p\in A\} \gg x(\log\log x)/(\log x)^2. \]

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arXiv2025

Multiplicative recurrence of Möbius transformations

We establish a complete characterization of multiplicative recurrence for images of the positive integers under Möbius transformations, answering a question of Donoso--Le--Moreira--Sun in the negative. As a consequence, we strengthen and extend a Diophantine approximation result of Charamaras--Mountakis--Tsinas, confirming their conjectures.

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arXiv2024

Zeros near $s=1$ and the constant term of $L'/L$ for $L$-functions in the Selberg class

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: $\bullet$ If $\mathcal{L}\in\mathcal{F}$ has no zeros $β+ iγ$ with $β> 1 - δ(\log q_{\mathcal{L}})^{-1}$, $|γ| < (\log q_{\mathcal{L}})^{-1/2}$ for some absolute $δ>0$, then $\Re(\ell_0(\mathcal{L})) \ll_{\mathcal{F}} \log q_{\mathcal{L}}$; $\bullet$ If $\Re(\ell_0(\mathcal{L})) \ll \log q_{\mathcal{L}}$ for all $\mathcal{L}\in \mathcal{F}$, then there is some absolute $δ> 0$ such that $\mathcal{L}$ has no zeros $β+ iγ$ with $β> 1 - δ(\log q_{\mathcal{L}})^{-1}$, $|γ| < (1-β)^{1/2}(\log q_{\mathcal{L}})^{-1/2}$. This generalizes, for instance, the case of families of Dedekind zeta functions of number fields with bounded degree.

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arXiv2024

On the size and structure of $t$-representable sumsets

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 size and 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 $h\ll m\ell t^{1/\ell}$.

math.CO↗

arXiv2024

Knights are 24/13 times faster than the king

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 \begin{equation*} \begin{aligned} \mathrm{N}_{a,b} = \{(a,b), (b,a), (-a,b), (-b,a), (-b,-a), (-a,-b), (a,-b), (b, -a)\},\newline \mathrm{K}=\{(1,0), (1,1), (0,1), (-1,1), (-1,0), (-1,-1), (0,-1), (1,-1)\} \subseteq \mathbb{Z}^2, \end{aligned} \end{equation*} 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 $2(a+b)b^2/(a^2+3b^2)$.

math.CO↗

arXiv2024

Classification of the conjugacy classes of $\widetilde{\mathrm{SL}}(2,\mathbb{R})$

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 $α\in \widetilde{\mathrm{SL}}_2(\mathbb{R})$, we show that its conjugacy class may be determined by three invariants: (i) Trace: the trace (valued in the set of positive real numbers $\mathbb{R}_{+}$) of its image $\overlineα$ in $\mathrm{PSL}_2(\mathbb{R})$; (ii) 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 $\mathrm{PSL}_2(\mathbb{R}) \curvearrowright \mathbb{S}^{1}$; (iii) The function $\ell^{\sharp}$: a conjugacy invariant length function introduced by S. Mochizuki [Res. Math. Sci. 3 (2016), 3:6].

math.GR↗

arXiv2021

On Landau-Siegel zeros and heights of singular moduli

Let $χ_D$ 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 \[ \frac{L'}{L}(1,χ_D) = \frac{1}{6}\, \mathrm{height}(j(τ_D)) - \frac{1}{2}\log|D| + C + o_{D\to -\infty}(1), \] where $τ_D = \frac 12(-δ+\sqrt{D})$ for $D \equiv δ~(\mathrm{mod}~4)$ and $j(\cdot)$ is the $j$-invariant function with $C = -1.057770\ldots$. Assuming the ``uniform'' $abc$-conjecture for number fields, we deduce that $L(β,χ_D)\ne 0$ with $β\geq 1 - \frac{\sqrt{5}φ+ o(1)}{\log|D|}$ where $φ= \frac{1+\sqrt{5}}{2}$, which we improve for smooth $D$.

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arXiv2019

An elementary heuristic for Hardy-Littlewood extended Goldbach's conjecture

The goal of this paper is to describe an elementary combinatorial heuristic that predicts Hardy and 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.

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