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Yifan Jing

Publications and source records attributed to Yifan Jing.

At least 19 recordsLinked to original sources
arXiv2026

A nonabelian Brunn-Minkowski inequality II

We prove that every unimodular locally compact group $G$ of noncompact Lie dimension $n$ satisfies the sharp Brunn--Minkowski inequality \[ \mu_G(XY)^{1/n}\ge\mu_G(X)^{1/n}+\mu_G(Y)^{1/n}, \] and establish a general form for arbitrary, possibly nonunimodular, locally compact groups. This fully confirms the nonabelian Brunn--Minkowski conjecture proposed by the present authors and Zhang. As an application, we obtain an isoperimetric inequality on symmetric spaces of noncompact type.

math.GR↗
arXiv2026

Near diagonal additive energy bound for points on algebraic surfaces

Let $F:\mathbb{R}^3\to\mathbb{R}$ be a polynomial that is irreducible over $\mathbb{R}$ with $\text{deg} F\geq2$. We prove that, for any finite $X\subset Z(F)$ that does not concentrate on affine lines, \[ E(X)=\#\{(a,b,c,d)\in X^4: a+b=c+d\}\ll_{\text{deg} F,\,\epsilon}(\# X)^{2+\epsilon}. \] In particular, this answers a question of Bourgain and Demeter concerning finite subsets of the unit sphere.

math.CA↗
arXiv2026

Kakeya sets and dimension compression in compact Lie groups

We study two Kakeya set problems in compact semisimple Lie groups. In the first, motivated by the group structure, a Kakeya set is required to contain a left coset of every closed one-dimensional torus. For a compact semisimple Lie group $G$ of dimension $d$ and rank $r$, we determine the optimal Minkowski dimension for this problem, proving that it is exactly $(d+r)/2$. The upper bound is obtained from a Lie-theoretic construction associated with a Chevalley involution, while the lower bound combines incidence geometry with geometry of numbers. We also consider a local Kakeya set problem, closer to the classical harmonic-analytic formulation, in which one requires a fixed-length one-parameter arc in every direction. For this problem we obtain lower bound $(d+1)/2$ and $(d+2)/2$ for odd and even $r$, and upper bound $(d+r)/2$. We conjecture that the upper bound should be sharp.

math.CA↗
arXiv2026

Computing Tools for Translation-Invariant Total Orders

We introduce TITO_Explore, a software package for representing and computing with Translation-Invariant Total Orders (TITOs). We define a canonical window notation for TITOs and design and implement algorithms for several computational tasks involving them. The package normalizes the window notation of a given TITO into its canonical form, computes its inversion set, compares the weak order between two TITOs, and computes the join of two specified TITOs. Our weak order comparison algorithm operates by partitioning the inversion sets into disjoint subsets, thereby breaking down the comparison problem into evaluations of paired subsets. The join algorithm uses an edge-weighted directed graph to represent inversions and converts the problem of finding the join into a weighted path problem in the graph.

math.CO↗
arXiv2026

A note on quantitative stability in Hilbert spaces

We study stability theory in Hilbert spaces quantitatively. We prove that the inner product on the unit ball is $(k,\epsilon)$-stable for all $k\ge \exp(\pi/\epsilon)$, and it is not $(k,\epsilon)$-stable for $k\le \exp(\log 2/\epsilon)$, showing that the growth is necessarily exponential in $1/\epsilon$. We then analyze how stability scales under nonlinear connectives applied to the inner product. In particular, for power-type predicates $f(x,y)=\langle x,y\rangle_+^\beta$ with $\beta<1$ we obtain upper and lower bounds of the form $\exp(C\epsilon^{-1/\beta})$, and for $\beta>1$ and integer powers $\langle x,y\rangle^d$ we retain the bilinear scale $\exp(C/\epsilon)$.

math.LO↗
arXiv2026

A structure theorem for sets with doubling $4+δ$

We prove a structural result for sets of integers with doubling at most $4 + δ$, with $δ>0$ sufficiently small. This generalises earlier work of Eberhard--Green--Manners which dealt with sets of integers with doubling strictly less than $4$, and makes progress towards a question of Green.

math.NT↗
arXiv2026

Roth-type theorems in $K_{s,t}$-free sets

We show that for all integers $2\le s\le t$, any $K_{s,t}$-free subset of $[N]$ with size $Ω(n^{1-1/s})$ must contain a nontrivial solution to every fixed translation-invariant linear equation in at least five variables. This extends earlier results for Sidon sets due to Conlon-Fox-Sudakov-Zhao and Prendiville to the full family of $K_{s,t}$-free sets. We also study the corresponding problem in vector spaces over finite fields. In $\mathbb F_q^n$ we obtain stronger quantitative bounds, including polylogarithmic savings, by combining Fourier-analytic transference with polynomial-method input from the arithmetic cycle-removal lemma of Fox-Lovász-Sauermann.

math.CO↗
arXiv2024

A group-action Szemerédi-Trotter theorem and applications to orchard problems in all characteristics

We establish a group-action version of the Szemerédi-Trotter theorem over any field, extending Bourgain's result for the group $\mathrm{SL}_2(k)$. As an Elekes-Szabó-type application, we obtain quantitative bounds on the number of collinear triples on reducible cubic surfaces in $\mathbb{P}^3(k)$, where $k = \mathbb{F}_{q}$ and $k = \mathbb{C}$, thereby improving a recent result by Bays, Dobrowolski, and the second author.

math.CO↗
arXiv2024

Brunn-Minkowski type estimates for certain discrete sumsets

Let $d,k$ be natural numbers and let $\mathcal{L}_1, \dots, \mathcal{L}_k \in \mathrm{GL}_d(\mathbb{Q})$ be linear transformations such that there are no non-trivial subspaces $U, V \subseteq \mathbb{Q}^d$ of the same dimension satisfying $\mathcal{L}_i(U) \subseteq V$ for every $1 \leq i \leq k$. For every non-empty, finite set $A \subset \mathbb{R}^d$, we prove that \[ |\mathcal{L}_1(A) + \dots + \mathcal{L}_k(A) | \geq k^d |A| - O_{d,k}(|A|^{1- δ}), \] where $δ>0$ is some absolute constant depending on $d,k$. Building on work of Conlon-Lim, we can show stronger lower bounds when $k$ is even and $\mathcal{L}_1, \dots, \mathcal{L}_k$ satisfy some further incongruence conditions, consequently resolving various cases of a conjecture of Bukh. Moreover, given any $d, k\in \mathbb{N}$ and any finite, non-empty set $A \subset \mathbb{R}^d$ not contained in a translate of some hyperplane, we prove sharp lower bounds for the cardinality of the $k$-fold sumset $kA$ in terms of $d,k$ and $|A|$. This can be seen as a $k$-fold generalisation of Freiman's lemma.

math.CO↗
arXiv2024

Efficient polynomial-time approximation scheme for the genus of dense graphs

The main results of this paper provide an Efficient Polynomial-Time Approximation Scheme (EPTAS) for approximating the genus (and non-orientable genus) of dense graphs. By dense we mean that $|E(G)|\ge α|V(G)|^2$ for some fixed $α>0$. While a constant factor approximation is trivial for this class of graphs, approximations with factor arbitrarily close to 1 need a sophisticated algorithm and complicated mathematical justification. More precisely, we provide an algorithm that for a given (dense) graph $G$ of order $n$ and given $\varepsilon>0$, returns an integer $g$ such that $G$ has an embedding into a surface of genus $g$, and this is $\varepsilon$-close to a minimum genus embedding in the sense that the minimum genus $\mathsf{g}(G)$ of $G$ satisfies: $\mathsf{g}(G)\le g\le (1+\varepsilon)\mathsf{g}(G)$. The running time of the algorithm is $O(f(\varepsilon)\,n^2)$, where $f(\cdot)$ is an explicit function. Next, we extend this algorithm to also output an embedding (rotation system) whose genus is $g$. This second algorithm is an Efficient Polynomial-time Randomized Approximation Scheme (EPRAS) and runs in time $O(f_1(\varepsilon)\,n^2)$.

math.CO↗
arXiv2024

Finding large additive and multiplicative Sidon sets in sets of integers

Given $h,g \in \mathbb{N}$, we write a set $X \subset \mathbb{Z}$ to be a $B_{h}^{+}[g]$ set if for any $n \in \mathbb{Z}$, the number of solutions to the additive equation $n = x_1 + \dots + x_h$ with $x_1, \dots, x_h \in X$ is at most $g$, where we consider two such solutions to be the same if they differ only in the ordering of the summands. We define a multiplicative $B_{h}^{\times}[g]$ set analogously. In this paper, we prove, amongst other results, that there exist absolute constants $g \in \mathbb{N}$ and $δ>0$ such that for any $h \in \mathbb{N}$ and for any finite set $A$ of integers, the largest $B_{h}^{+}[g]$ set $B$ inside $A$ and the largest $B_{h}^{\times}[g]$ set $C$ inside $A$ satisfy \[ \max \{ |B| , |C| \} \gg_{h} |A|^{(1+ δ)/h }. \] In fact, when $h=2$, we may set $g = 31$, and when $h$ is sufficiently large, we may set $g = 1$ and $δ\gg (\log \log h)^{1/2 - o(1)}$. The former makes progress towards a recent conjecture of Klurman--Pohoata and quantitatively strengthens previous work of Shkredov.

math.NT↗
arXiv2023

A note on the largest sum-free sets of integers

Given $A$ a set of $N$ positive integers, an old question in additive combinatorics asks that whether $A$ contains a sum-free subset of size at least $N/3+ω(N)$ for some increasing unbounded function $ω$. The question is generally attacked in the literature by considering another conjecture, which asserts that as $N\to\infty$, $\max_{x\in\mathbb{R}/\mathbb{Z}}\sum_{n\in A}({\bf 1}_{(1/3,2/3)}-1/3)(nx)\to\infty$. This conjecture, if true, would also imply that a similar phenomenon occurs for $(2k,4k)$-sum-free sets for every $k\geq1$. In this note, we prove the latter result directly. The new ingredient of our proof is a structural analysis on the host set $A$, which might be of independent interest.

math.CO↗
arXiv2023

Kemperman's inequality and Freiman's lemma via few translates

Let $G$ be a connected compact group equipped with the normalised Haar measure $μ$. Our first result shows that given $α, β>0$, there is a constant $c = c(α,β)>0$ such that for any compact sets $A,B\subseteq G$ with $ αμ(B)\geqμ(A)\geq μ(B) $ and $ μ(A)+μ(B)\leq 1-β$, there exist $b_1,\dots b_c\in B$ such that \[ μ(A\cdot \{b_1,\dots,b_c\})\geq μ(A)+μ(B).\] A special case of this, that is, when $G=\mathbb{T}^d$, confirms a recent conjecture of Bollobás, Leader and Tiba. We also prove a quantitatively stronger version of such a result in the discrete setting of $\mathbb{R}^d$. Thus, given $d \in \mathbb{N}$, we show that there exists $c = c(d) >0$ such that for any finite, non-empty set $A \subseteq \mathbb{R}^d$ which is not contained in a translate of a hyperplane, one can find $a_1, \dots, a_c \in A$ satisfying \[ |A+ \{a_1, \dots, a_c\}| \geq (d+1)|A| - O_d(1). \] The main term here is optimal and recovers the bounds given by Freiman's lemma up to the $O_d(1)$ error term.

math.CO↗
arXiv2023

A nonabelian Brunn-Minkowski inequality

Henstock and Macbeath asked in 1953 whether the Brunn-Minkowski inequality can be generalized to nonabelian locally compact groups; questions along the same line were also asked by Hrushovski, McCrudden, and Tao. We obtain here such an inequality and prove that it is sharp for helix-free locally compact groups, which includes real linear algebraic groups, Nash groups, semisimple Lie groups with finite center, solvable Lie groups, etc. The proof follows an induction on dimension strategy; new ingredients include an understanding of the role played by maximal compact subgroups of Lie groups, a necessary modified form of the inequality which is also applicable to nonunimodular locally compact groups, and a proportionated averaging trick.

math.GR↗
arXiv2023

Measure doubling of small sets in $\mathrm{SO}(3,\mathbb{R})$

Let $\mathrm{SO}(3,\mathbb{R})$ be the 3D-rotation group equipped with the real-manifold topology and the normalized Haar measure $μ$. Resolving a problem by Breuillard and Green, we show that if $A \subseteq \mathrm{SO}(3,\mathbb{R})$ is an open subset with sufficiently small measure, then $$ μ(A^2) > 3.99 μ(A).$$ We also show a more general result for the product of two sets, which can be seen as a Brunn-Minkowski-type inequality for sets with small measure in $\mathrm{SO}(3,\mathbb{R})$.

math.GR↗
arXiv2023

Integer colorings with forbidden rainbow sums

For a set of positive integers $A \subseteq [n]$, an $r$-coloring of $A$ is rainbow sum-free if it contains no rainbow Schur triple. In this paper we initiate the study of the rainbow Erdős-Rothchild problem in the context of sum-free sets, which asks for the subsets of $[n]$ with the maximum number of rainbow sum-free $r$-colorings. We show that for $r=3$, the interval $[n]$ is optimal, while for $r\geq8$, the set $[\lfloor n/2 \rfloor, n]$ is optimal. We also prove a stability theorem for $r\geq4$. The proofs rely on the hypergraph container method, and some ad-hoc stability analysis.

math.CO↗
arXiv2023

Measure growth in compact semisimple Lie groups and the Kemperman Inverse Problem

Suppose $G$ is a compact semisimple Lie group, $μ$ is the normalized Haar measure on $G$, and $A, A^2 \subseteq G$ are measurable. We show that $$μ(A^2)\geq \min\{1, 2μ(A)+ημ(A)(1-2μ(A))\}$$ with the absolute constant $η>0$ (independent from the choice of $G$) quantitatively determined. We also show a more general result for connected compact groups without a toric quotient and resolve the Kemperman Inverse Problem from 1964.

math.GR↗