arXiv subjects

Joseph Harrison

Publications and source records attributed to Joseph Harrison.

3 recordsLinked to original sources

arXiv2026

Parameterwise Sharpness of Khovanskii's Bezout-type Bound for Pfaffian Functions

Khovanskii's theorem gives a Bezout-type upper bound for the number of isolated real solutions of a system of $n$ Pfaffian equations in $n$ variables in terms of three complexity parameters: the chain-degree $\alpha$, the degrees $\beta_i$ of the Pfaffian functions, and the order $s$ of the underlying Pfaffian chain. Despite its fundamental role in Pfaffian geometry and o-minimality, little is known about the sharpness of this bound. We investigate the theorem from a parameter-by-parameter perspective. We show that its dependence on the chain-degree $\alpha$ is asymptotically sharp by constructing, for every $\alpha,s \in \mathbb{N}$, a Pfaffian function of format $(\alpha,1,s)$ with at least $\alpha^s$ nondegenerate real zeros. We also show that its dependence on the degrees $\beta_i$ is asymptotically sharp: for fixed $n$ and $s$, we construct Pfaffian systems having $\Omega_{n,s}(\beta^{n+s})$ regular common zeros, matching the order of growth predicted by Khovanskii's theorem as $\beta\to\infty$.

math.AG↗

arXiv2026

Uniform sum-product phenomenon for algebraic groups and Bremner's conjecture

In this paper we combine methods from additive combinatorics and Diophantine geometry to study the generalised sum-product phenomenon in algebraic groups. As an application of this circle of ideas, we resolve a conjecture of Bremner on arithmetic progressions in coordinates of elliptic curves, along with various other generalisations studied in the literature. We also prove a uniform Bourgain--Chang-type sum-product estimate for general $1$-dimensional algebraic groups $G$ over $\mathbb{C}$. Using these ideas, we provide an alternative solution to a problem of Bays--Breuillard. Furthermore, we show an Elekes--Szab\'{o} type result in the same setting for sets with small doubling, improving upon an earlier result of Bays--Breuillard when $G$ is not $\mathbb{G}_a$. Our power saving here can be shown to be quantitatively optimal. We use a combination of deep, classical results in Diophantine geometry due to David--Philippon, Laurent and Evertse--Schmidt--Schlickewei along with the recent breakthrough work on the weak Polynomial Freiman--Ruzsa conjecture over integers due to Gowers--Green--Manners--Tao.

math.NT↗

arXiv2025

Additive relations in irrational powers

We investigate the interaction between raising to an irrational power and addition of real numbers. Thus, for a finite set $A$ of non-negative real numbers, let $A^{[c]} = \{a^c : a \in A\}$. When $k$ is a positive integer, $c$ is a real irrational number, and $A$ is a subset of an $N$-term arithmetic progression in $\mathbb{R}_{\geq 0}$ having cardinality at least a power of $\log{N}$, we prove that the $k$-fold sumset $|kA^{[c]}| \sim_k |A|^k/k!$ as $|A| \to \infty$. This result is uniform in $c$. When $A = \{1, \dots, N\}$ and $k = 2$, this result can be combined with existing works to show that $|A^{[c]} + A^{[c]}| \sim N^2/2$ as $N \to \infty$ whenever $c \in \mathbb{R} \setminus \{0, 1, 2\}$. The sumset lower bound follows from a bound on the number of equal sums of $r$ and $s \geq r$ elements of $A^{[c]}$ (by taking $r = s = k$). When $r = s = 2$ or $s > r$, our bound is optimal up to a power of $\log N$. This bound is proved using a functional transcendence theorem for certain endomorphisms of $\mathbb{R}_{>0}^n$, and innovations in the Pila--Wilkie counting theorem in $\mathbb{R}_{\exp}$ due to Binyamini, Novikov and Zak. In a different direction, we provide a Diophantine approximation criterion on $c$ that, when satisfied, ensures that a linear form in the $c$-th powers of multiplicatively independent integers does not vanish. The proof involves linear forms in logarithms. This provides a new proof of a fact, due to Bays--Kirby--Wilkie and Jones--Servi, that when $A$ is a multiplicatively independent set of positive integers, there are infinitely many effectively computable real numbers $c$ such that $A^{[c]}$ is linearly independent over $\mathbb{Q}$.

math.NT↗