arXiv subjects
Publications and source records attributed to Chieu-Minh Tran.
These open problems were presented in the Problem Sessions held during the Tianyuan Workshop on Definability and Computation, June 22-26, 2026. The problems are organized into sections named after their contributors, in the order of their presentations during the workshop. Notes were taken and compiled by Wei Dai, Xiangxi Hu, Yingying Jiang, Ruiwen Li, Tianhao Wang, Xu Wang, and Jie Zou.
We study multipartite hypergraphs definable in $1$-semi-equational theories and prove almost-linear Zarankiewicz bounds in every fixed arity $r\geq2$. If $T$ is a $1$-semi-equational theory, then, for every formula $\varphi$ and fixed $t,r\geq2$, there is a constant $c$ such that each $K_{t,\ldots,t}$-free $r$-partite hypergraph defined by $\varphi$ on $n$ vertices has $O_{T,\varphi,t,r}\!\left( n^{r-1}(1+\log(1+n))^c \right) $ edges. Put $\alpha_k=\min\{k-1,2\}$. In the bipartite case, a Boolean combination of $m$ $(k,1)$-semi-equations has $ O_{k,t,m}\!\left( n(1+\log(1+n))^{(m-1)\alpha_k} \right) $ edges whenever it is $K_{t,t}$-free. In particular, a relation defined by one $(k,1)$-semi-equation or its negation has a linear bound. The proofs combine incidence estimates for indexed set systems with low-crossing orderings of finite $k$-wise laminar families. Consequently, no $1$-semi-equational theory locally trace-defines an infinite domain.
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.
We establish absolute and relative almost-linear Zarankiewicz bounds for semilinear relations in valued vector spaces. For every fixed arity and description complexity, a $K_{t,\ldots,t}$-free semilinear $r$-partite hypergraph has at most \[ O\!\left(n^{r-1}(\log n)^c\right) \] edges, where $c$ depends only on the arity and the number of valuative literals. In the bipartite case a separate arbitrary-trace argument gives the explicit bound $O(n(\log n)^{2s})$ for description complexity $(\rho,s)$. We also prove a relative extension theorem: intersecting any relation with a hereditary almost-linear profile by $s$ affine moving-radius comparisons increases the logarithmic exponent by at most $2s$. For the additive affine-valuative structures on $\mathbb Q_p$ and $\mathbb C_p$, quantifier elimination converts these semilinear results into bounds for all definable relations. Finally, over every valued field with infinite value group, we construct $K_{2,2}$-free semilinear point--box graphs of description complexity $(1,4)$ with $\Omega(n\log n/\log\log n)$ edges.
We prove an analogue of Morley's categoricity theorem where cardinality is replaced by the recursion-theoretic notion of arithmetic degree. We say that a complete arithmetically definable theory $T$ is $D$-categorical if any two arithmetically extendible models of $T$ of arithmetic degree $D$, considered over a common elementary submodel with arithmetical elementary diagram, are isomorphic over that submodel by an isomorphism which preserves the complexity of sets of degree $D$. Here an arithmetically extendible model means an elementary substructure of a model whose elementary diagram is arithmetical. Our main result is: If $T$ is $D_1$-categorical for some nonzero arithmetic degree $D_1$, then $T$ is $D_2$-categorical for every nonzero arithmetic degree $D_2$. We also show that, assuming ZFC, $D$-categoricity for some nonzero arithmetic degree is equivalent to uncountable categoricity.
Fix a field $K$. We show that $K$ is large if and only if some elementary extension of $K$ is the fraction field of a henselian local domain which is not a field. The proof uses a new result about the \'etale-open topology over $K$: if $K$ is not separably closed and $V \to W$ is an \'etale morphism of $K$-varieties then $V(K) \to W(K)$ is a local homeomorphism in the \'etale-open topology. This, in turn, follows from results comparing the \'etale-open topology on $V(K)$ and the finite-closed topology on $V(K)$, newly introduced in this paper. We show that the \'etale-open topology refines the finite-closed topology when $K$ is perfect, and that the finite-closed topology refines the \'etale-open topology when $K$ is bounded. It follows that these two topologies agree in many natural examples. On the other hand, we construct several examples where these two differ, which allows us to answer a question of Lampe.
We show that the shatter function of a semilinear set system on $\mathbb{R}^m$ is asymptotic to a polynomial. This confirms, for the structure $(\mathbb{R}; +, <)$, a conjecture of Chernikov and is a step towards characterizing model-theoretic linearity via shatter functions.
We show that if G is a split semisimple algebraic group over a model complete field K, then the groups G(K) and G(K)' (the commutator group which is a ``Chevalley group'' as for example the group PSL_2(K)) are model complete as well.
We consider a (possibly discrete) unimodular locally compact group $G$ with Haar measure $μ_G$, and a compact $A\subseteq G$ of positive measure with $μ_G(A^2)\leq Kμ_G(A)$. Let $H$ be a closed normal subgroup of G and $π: G \rightarrow G/H$ be the quotient map. With the further assumption that $A= A^{-1}$, we show $$μ_{G/H}(πA ^2) \leq K^2 μ_{G/H}(πA).$$ We also demonstrate that $K^2$ cannot be replaced by $(1-ε)K^2$ for any $ε>0$. In the general case (without $A=A^{-1}$), we show $μ_{G/H}(πA ^2) \leq K^3 μ_{G/H}(πA)$, improving an earlier result by An, Jing, Zhang, and the third author. Moreover, we are able to extract a compact set $B\subseteq A$ with $μ_G(B)> μ_G(A)/2$ such that $ μ_{G/H}(πB^2) < 2K μ_{G/H}(πB)$.
We generalize the Cauchy-Davenport theorem to locally compact groups.
Let $\mathbb C$ be the set of complex numbers, and let $\mathcal P$ be a collection of complex polynomial maps in several variables. Assuming at least one $P\in\mathcal P$ depends on at least two variables, we classify all possibilities for the structure $(\mathscr M;\mathcal P)$ up to definable equivalence. In particular, outside a short list of exceptions, we show that $(\mathscr M;\mathcal P)$ always defines $+$ and $\times$. Our tools include Zilber's Restricted Trichotomy, as well as the classification of symmetric non-expanding pairs of polynomials over $\mathbb C$ from arithmetic combinatorics. Along the way, we also give a new condition for a reduct $\mathscr M=(M,...)$ of a smooth curve over an algebraically closed field to recover all constructible subsets of powers of $M$.
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.
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 $\mathrm{SO}(3,\mathbb{R})$.
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.
For a bivariate $P(x,y) \in \mathbb{R}[x,y]\setminus (\mathbb{R}[x] \cup \mathbb{R}[y])$, our first result shows that for all finite $A \subseteq \mathbb{R}$, $|P(A,A)|\geq α|A|^{5/4}$ with $α=α(\mathrm{deg} P) \in \mathbb{R}^{>0}$ unless $$ P(x,y)=f(γu(x)+δu(y)) \text{ or } P(x,y)=f(u^m(x)u^n(y)) $$ for some univariate $f, u \in \mathbb{R}[t]\setminus \mathbb{R}$, constants $γ, δ\in \mathbb{R}^{\neq 0}$, and $m, n\in \mathbb{N}^{\geq 1}$. This resolves the symmetric nonexpanders classification problem proposed by de Zeeuw. Our second and third results are sum-product type theorems for two polynomials, generalizing the classical result by Erdos and Szemerédi as well as a theorem by Shen. We also obtained similar results for $\mathbb{C}$, and from this deduce results for fields of characteristic $0$ and fields of large prime characteristic. The proofs of our results use tools from semialgebraic/o-minimal geometry.
Suppose $G$ is a connected noncompact locally compact group, $A,B$ are nonempty and compact subsets of $G$, $μ$ is a left Haar measure on $G$. Assuming that $G$ is unimodular, and $ μ(A^2) < K μ(A) $ with $K>1$ a fixed constant, our first result shows that there is a continuous surjective group homomorphism $χ: G\to L$ with compact kernel, where $L$ is a Lie group with $$\dim(L) \leq \lfloor\log K\rfloor(\lfloor\log K\rfloor+1)/2.$$ We also demonstrate that this dimension bound is sharp, establish the relationship between $A$ and its image under the quotient map, and obtain a more general version of this result for the product set $AB$ without assuming unimodularity. Our second result classifies $G,A,B$ where $A,B$ have nearly minimal expansions (when $G$ is unimodular, this just means $μ(AB)$ is close to $μ(A)+μ(B)$). This answers a question suggested by Griesmer and Tao, and completes the last open case of the inverse Kemperman problem. The proofs of both results involve a new analysis of locally compact group $G$ with bounded $n-h$, where $n-h$ is an invariant of $G$ appearing in the recently developed nonabelian Brunn-Minkowski inequality. We also generalize Ruzsa's distance and related results to possibly nonunimodular locally compact groups.
We classify the stable formulas in the theory of Dense Linear Orders without endpoints, the stable formulas in the theory of Divisible Abelian Groups, and the stable formulas without parameters in the theory of Real Closed Fields. The third result, unexpectedly, requires the Hironaka's theorem on resolution of singularities.
For an arbitrary field $K$ and $K$-variety $V$, we introduce the étale-open topology on the set $V(K)$ of $K$-points of $V$. This topology agrees with the Zariski topology, Euclidean topology, or valuation topology when $K$ is separably closed, real closed, or $p$-adically closed, respectively. Topological properties of the étale-open topology corresponds to algebraic properties of $K$. For example, the étale-open topology on $\mathbb{A}^1(K)$ is not discrete if and only if $K$ is large. As an application, we show that a large stable field is separably closed.