arXiv subjects

Jeck Lim

Publications and source records attributed to Jeck Lim.

15 recordsLinked to original sources

arXiv2026

Human agency in initial human-AI proof formalization workflows

For centuries, human mathematicians have written proofs to substantiate their mathematical arguments; yet, the ability to automatically verify the validity of proofs has long been a challenge. Advances in AI systems' ability to generate code and engage in increasingly high-level mathematical reasoning promise to transform people's ability to formalize and thereby verify proofs. While many works focus on benchmarking the current frontier, we instead study how people use these tools and apply agency in doing so. We conduct a mixed-methods analysis into the initial impact of AI on people's formalization workflows: what people claim they want, what they see as the barriers to those visions, and how they actually use and adapt AI in practice. A qualitative survey reveals that people's preferences are diverse, but with a general desire for AI assistance in formalization that preserves high-level human control and agency over the proof discovery process. To assess how people actually engage with AI for formalization, we conduct a controlled user study in which participants formalize informal math problems and their proofs, with and without AI, across a range of mathematical problems at varying levels of difficulty and domains. Despite limitations of the tools at the time for autoformalization, participants tended to attain higher formalization accuracy when allowed access to AI tools than when formalizing on their own, with most participants flexibly choosing to use multiple different AI tools. Taken together, our work sheds light on the early stages of AI integration into formalization workflows, involving an intimate interplay of human agency and AI engagement.

cs.AI↗

arXiv2026

On the largest sum-free subset of the lattice cube

We determine the limiting density of the largest sum-free subset of the lattice cube $\{1,2,\ldots,n\}^d$ for all $d$, thus resolving the natural conjecture that it is constructed by two appropriate hyperplane slices. Equivalently, we show that the largest measure of a sum-free subset of the hypercube $(0,1)^d\subset \mathbb{R}^d$ is attained by $\setcond{x\in (0,1)^d}{1\leq L(x)<2}$ for some linear map $L:\mathbb{R}^d\to \mathbb{R}$. It is natural to conjecture that the same phenomenon might hold if one replaces the hypercube by any convex set not containing the origin, but we give an example to show that for sufficiently large $d$ this is not the case.

math.CO↗

arXiv2025

On exponential Freiman dimension

The exponential Freiman dimension of a finite set $A \subset \mathbb{R}^{m}$, introduced by Green and Tao in 2006, represents the largest positive integer $d$ for which $A$ contains the vertices of a non-degenerate $d$-dimensional parallelepiped. For every $d \geq 1$, we precisely determine the largest constant $C_{d}>0$ (exponential in $d$) for which $$|A+A| \geq C_{d}|A| - O_{d}(1)$$ holds for all sets $A$ with exponential Freiman dimension $d$.

math.CO↗

arXiv2025

Balancing games on unbounded sets

For a finite set $V\subset \mathbb{R}^n$, a set $T\subset \mathbb{R}^n$ is called $V$-closed if $t \in T$ and $v\in V$ imply that either $t+v\in T$ or $t-v \in T$. The set $P(V):=\{\sum_{v \in W} v: W \subset V\}$ is clearly $V$-closed and so are its translates. We show, assuming $V$ contains no parallel vectors, that if $T$ is closed and $V$-closed, and $x \in T$ is an extreme point of $\operatorname{cl} \operatorname{conv} T$, then there is a translate of $P(V)$ containing $x$ and contained in $\operatorname{conv} T$. This result is used to determine the value of a special balancing game. A byproduct is that when $m\ge 2$ and is not a power of 2, then the $m$-sets of a $2m$-set can be coloured Red and Blue so that complementary $m$-sets have distinct colours and every point of the $2m$-set is contained in the same number of Red and Blue sets.

math.CO↗

arXiv2025

Sums of algebraic dilates

We show that if $λ_1,\ldots,λ_k$ are algebraic numbers, then $$|A+λ_1\cdot A+\dots+λ_k\cdot A|\geq H(λ_1,\ldots,λ_k)|A|-o(|A|)$$ for all finite subsets $A$ of $\mathbb{C}$, where $H(λ_1,\ldots,λ_k)$ is an explicit constant that is best possible. The proof combines several ingredients, including a lower bound estimate on the measure of sums of linear transformations of compact sets in $\mathbb{R}^d$, a variant of Freiman's theorem tuned specifically to sums of dilates and the analysis of what we call lattice density, which succinctly captures how a subset of $\mathbb{Z}^d$ is arranged relative to a given flag of lattices. As an application, we revisit the study of sums of linear transformations of finite sets, in particular proving an asymptotically best possible lower bound for sums of two linear transformations.

math.CO↗

arXiv2025

A Purely Geometric Variant of the Gale-Berlekamp Switching Game

We introduce the following variant of the Gale-Berlekamp switching game. Let $P$ be a set of n noncollinear points in the plane, each of them having weight $+1$ or $-1$. At each step, we pick a line $\ell$ passing through at least two points of $P$, and switch the sign of every point $p \in P\cap\ell$. The objective is to maximize the total weight of the elements of $P$. We show that one can always achieve that this quantity is at least $n - o(n)$, as $n\rightarrow\infty$, and at least $n/3$, for every $n$. Moreover, these can be attained by a polynomial time algorithm.

cs.CG↗

arXiv2025

Evasive sets, twisted varieties, and container-clique trees

In the affine space $\mathbb{F}_q^n$ over the finite field of order $q$, a point set $S$ is said to be $(d,k,r)$-evasive if the intersection between $S$ and any variety, of dimension $k$ and degree at most $d$, has cardinality less than $r$. As $q$ tends to infinity, the size of a $(d,k,r)$-evasive set in $\mathbb{F}_q^n$ is at most $O\left(q^{n-k}\right)$ by a simple averaging argument. We exhibit the existence of such evasive sets of sizes at least $Ω\left(q^{n-k}\right)$ for much smaller values of $r$ than previously known constructions, and establish an enumerative upper bound $2^{O(q^{n-k})}$ for the total number of such evasive sets. The existence result is based on our study of twisted varieties. In the projective space $\mathbb{P}^n$ over an algebraically closed field, a variety $V$ is said to be $d$-twisted if the intersection between $V$ and any variety, of dimension $n - \dim(V)$ and degree at most $d$, has dimension zero. We prove an upper bound on the smallest possible degree of twisted varieties which is best possible in a mild sense. The enumeration result includes a new technique for the container method which we believe is of independent interest. To illustrate the potential of this technique, we give a simpler proof of a result by Chen--Liu--Nie--Zeng that characterizes the maximum size of a collinear-triple-free subset in a random sampling of $ \mathbb{F}_q^2$ up to polylogarithmic factors.

math.CO↗

arXiv2025

Everywhere unbalanced configurations

An old problem in discrete geometry, originating with Kupitz, asks whether there is a fixed natural number $k$ such that every finite set of points in the plane has a line through at least two of its points where the number of points on either side of this line differ by at most $k$. We give a negative answer to a natural variant of this problem, showing that for every natural number $k$ there exists a finite set of points in the plane together with a pseudoline arrangement such that each pseudoline contains at least two points and there is a pseudoline through any pair of points where the number of points on either side of each pseudoline differ by at least $k$. Moreover, we may find such a configuration with at most $2^{2^{ck}}$ points, which, by a result of Pinchasi, is best possible up to the value of the constant $c$.

math.CO↗

arXiv2024

Sums of linear transformations

We show that if $\mathcal{L}_1$ and $\mathcal{L}_2$ are linear transformations from $\mathbb{Z}^d$ to $\mathbb{Z}^d$ satisfying certain mild conditions, then, for any finite subset $A$ of $\mathbb{Z}^d$, $$|\mathcal{L}_1 A+\mathcal{L}_2 A|\geq \left(|\det(\mathcal{L}_1)|^{1/d}+|\det(\mathcal{L}_2)|^{1/d}\right)^d|A|- o(|A|).$$ This result corrects and confirms the two-summand case of a conjecture of Bukh and is best possible up to the lower-order term for certain choices of $\mathcal{L}_1$ and $\mathcal{L}_2$. As an application, we prove a lower bound for $|A + λ\cdot A|$ when $A$ is a finite set of real numbers and $λ$ is an algebraic number. In particular, when $λ$ is of the form $(p/q)^{1/d}$ for some $p, q, d \in \mathbb{N}$, each taken as small as possible for such a representation, we show that $$|A + λ\cdot A| \geq (p^{1/d} + q^{1/d})^d |A| - o(|A|).$$ This is again best possible up to the lower-order term and extends a recent result of Krachun and Petrov which treated the case $λ= \sqrt{2}$.

math.CO↗

arXiv2024

Sums of dilates over groups of prime order

For $p$ prime, $A \subseteq \mathbb{Z}/p\mathbb{Z}$ and $λ\in \mathbb{Z}$, the sum of dilates $A + λ\cdot A$ is defined by \[A + λ\cdot A = \{a + λa' : a, a' \in A\}.\] The basic problem on such sums of dilates asks for the minimum size of $|A + λ\cdot A|$ for given $λ$, $A$ of given density $α$, and $p$ tending to infinity. We investigate this problem for $α$ fixed and $λ$ tending to infinity, proving near-optimal bounds in this case.

math.CO↗

arXiv2024

On differences of two harmonic numbers

We prove that the existence of infinitely many $(m_k, n_k) \in \mathbb{N}^2$ such that the difference of harmonic numbers $H_{m_k} - H_{n_k}$ approximates 1 well $$ \lim_{k \rightarrow \infty} \left| \sum_{\ell = n}^{m_k} \frac{1}{\ell} - 1 \right|\cdot n_k^2 = 0.$$ This answers a question of Erdős and Graham. The construction uses asymptotics for harmonic numbers, the precise nature of the continued fraction expansion of $e$ and a suitable rescaling of a subsequence of convergents. We also prove a quantitative rate by appealing to techniques of Heilbronn, Danicic, Harman, Hooley and others regarding $\min_{1 \leq n \leq N} \min_{m \in \mathbb{N}}\| n^2 θ- m\|$.

math.CO↗

arXiv2023

Difference sets in $\mathbb{R}^d$

Let $d \geq 2$ be a natural number. We show that $$|A-A| \geq \left(2d-2 + \frac{1}{d-1}\right)|A|-(2d^2-4d+3)$$ for any sufficiently large finite subset $A$ of $\mathbb{R}^d$ that is not contained in a translate of a hyperplane. By a construction of Stanchescu, this is best possible and thus resolves an old question first raised by Uhrin.

math.CO↗

arXiv2023

Sums of transcendental dilates

We show that there is an absolute constant $c>0$ such that $|A+λ\cdot A|\geq e^{c\sqrt{\log |A|}}|A|$ for any finite subset $A$ of $\mathbb{R}$ and any transcendental number $λ\in\mathbb{R}$. By a construction of Konyagin and Laba, this is best possible up to the constant $c$.

math.CO↗

arXiv2022

Fixing a hole

We show that any finite $S \subset \mathbb{R}^d$ in general position has arbitrarily large supersets $T \supseteq S$ in general position with the property that $T$ contains no empty convex polygon, or hole, with $C_d$ points, where $C_d$ is an integer that depends only on the dimension $d$. This generalises results of Horton and Valtr which treat the case $S = \emptyset$. The key step in our proof, which may be of independent interest, is to show that there are arbitrarily small perturbations of the set of lattice points $[n]^d$ with no large holes.

math.CO↗

arXiv2020

A generalisation of Seymour's second neighbourhood conjecture

In this note we propose a generalisation of Seymour's Second Neighbourhood Conjecture to two directed graphs on a vertex set. We prove that this generalisation holds in the case of tournaments, and we show that a natural strengthening of this conjecture does not hold.

math.CO↗