arXiv subjects

Fredy Yip

Publications and source records attributed to Fredy Yip.

9 recordsLinked to original sources

arXiv2026

Product-free subsets of $(0,1)$

The third problem in Ben Green's collection of 100 open problems asks whether an open subset of $(0,1)$ that does not contain $x,y,z$ with $xy=z$ must have measure at most 1/3. We give an affirmative answer to this question. As part of the proof we obtain a result of independent interest that gives a lower bound for the size of the sumset and the difference set of a set of reals in terms not just of its size but also of a parameter that measures how far it is from being an interval.

math.CO↗

arXiv2026

Multivariate and quantitative Erd\H{o}s-Kac laws for Beatty sequences

The classical Erd\H{o}s-Kac theorem states that for $n$ chosen uniformly at random from $1, \dots, N$, the random variable $(\omega(n) - \log\log N)/\sqrt{\log\log N}$ converges in distribution to the standard Gaussian as $N$ tends to infinity. Banks and Shparlinski showed that this Gaussian convergence holds for any Beatty sequence $\lfloor\alpha n + \beta\rfloor$ in place of $n$. Continuing in this spirit, Crn\v{c}evi\'c, Hern\'andez, Rizk, Sereesuchart and Tao considered the joint distribution of $\omega(n)$ and $\omega(\lfloor\alpha n\rfloor)$, which they showed to be asymptotically independent for irrational values of $\alpha$. Generalising both results, we show that for any positive integer $k$, real numbers $\alpha_1, \dots, \alpha_k > 0$ and $\beta_1, \dots, \beta_k$, where $\alpha_i/\alpha_j$ is irrational for $i\neq j$, the joint distribution of $(\omega(\lfloor\alpha_in + \beta_i\rfloor) - \log\log N)/\sqrt{\log\log N}$ converges to the $k$-dimensional standard Gaussian. We next discuss quantitative bounds on the rate of convergence which do not depend on the values taken by the relevant parameters. Banks and Shparlinski remarked that such quantitative bounds may be given for a single Beatty sequence $\lfloor\alpha n + \beta\rfloor$ under Diophantine type assumptions on $\alpha$. We show that such assumptions are in fact unnecessary. Specifically, for any real numbers $\alpha > 0, \beta$, we show that the Kolmogorov distance between the random variable $(\omega(\lfloor\alpha n + \beta\rfloor) - \log\log N)/\sqrt{\log\log N}$ and the standard Gaussian is bounded above by $O(\log\log\log N/\sqrt{\log\log N})$ as $N$ tends to infinity. On the other hand, we show that universal quantitative bounds of this kind do not exist for higher-degree generalised polynomials or for the joint convergence for multiple Beatty sequences.

math.NT↗

arXiv2025

On a problem of Erd\H{o}s and Ingham

We give a short and elementary argument answering a question of Erd\H{o}s and Ingham negatively. Erd\H{o}s and Ingham showed that a Tauberian estimate they considered was equivalent to the non-vanishing of $1+\sum_{k}a_k^{-1-it}$ for any real number $t$ and any sequence $1<a_1<a_2<\cdots$ of positive integers such that $\sum_k a_k^{-1}<\infty$. We disprove this statement. In fact, we show that for any complex number $\lambda$ and any non-zero real number $t$, there exists a sequence $1<a_1<a_2<\cdots$ of positive integers such that $\sum_k a_k^{-1}<\infty$ and $\sum_k a_k^{-1-it} = \lambda$.

math.CA↗

arXiv2025

Combinatorial foundations for solvable chaotic local Euclidean quantum circuits in two dimensions

We investigate a graph-theoretic problem motivated by questions in quantum computing concerning the propagation of information in quantum circuits. A graph $G$ is said to be a bounded extension of its subgraph $L$ if they share the same vertex set, and the graph distance $d_L(u, v)$ is uniformly bounded for edges $uv\in G$. Given vertices $u, v$ in $G$ and an integer $k$, the geodesic slice $S(u, v, k)$ denotes the subset of vertices $w$ lying on a geodesic in $G$ between $u$ and $v$ with $d_G(u, w) = k$. We say that $G$ has bounded geodesic slices if $|S(u, v, k)|$ is uniformly bounded over all $u, v, k$. We call a graph $L$ geodesically directable if it has a bounded extension $G$ with bounded geodesic slices. Contrary to previous expectations, we prove that $\mathbb{Z}^2$ is geodesically directable. Physically, this provides a setting in which one could devise exactly-solvable chaotic local quantum circuits with non-trivial correlation patterns on 2D Euclidean lattices. In fact, we show that any bounded extension of $\mathbb{Z}^2$ is geodesically directable. This further implies that all two-dimensional regular tilings are geodesically directable.

quant-ph↗

arXiv2025

On the sum of the largest and smallest eigenvalues of graphs with high odd girth

The sum $\lambda_1 + \lambda_n$ of the maximum and minimum eigenvalues, and the odd girth of a graph both measure bipartiteness. We seek to relate these measures. In particular, for an odd integer $k\geq 3$, let $\gamma_k$ denote the supremum of $\frac{\lambda_1 + \lambda_n}{n}$ over graphs without odd cycles of length less than $k$. The example of the $k$-cycle $C_k$ shows that $\gamma_k\geq \Omega(k^{-3})$. In their recent work, Abiad, Taranchuk, and Van Veluw showed that $\gamma_k\leq O(k^{-1})$ and asked to determine the asymptotics of $\gamma_k$. Using approximation theory, we show that $\gamma_k\leq O(k^{-3}(\log k)^3)$, giving a tight upper bound up to a poly-logarithmic factor.

math.CO↗

arXiv2025

Short monochromatic odd cycles

It is easy to see that every $k$-edge-colouring of the complete graph on $2^k+1$ vertices contains a monochromatic odd cycle. In 1973, Erd\H{o}s and Graham asked to estimate the smallest $L(k)$ such that every $k$-edge-colouring of $K_{2^k+1}$ contains a monochromatic odd cycle of length at most $L(k)$. Recently, Gir\~ao and Hunter obtained the first nontrivial upper bound by showing that $L(k)=O(\frac{2^k}{k^{1-o(1)}})$, which improves the trivial bound by a polynomial factor. We obtain an exponential improvement by proving that $L(k)=O(k^{3/2}2^{k/2})$. Our proof combines tools from algebraic combinatorics and approximation theory.

math.CO↗

arXiv2024

A variant of the Erd\H{o}s-Gy\'arf\'as problem for $K_8$

Recently, Alon initiated the study of graph codes and their linear variants in analogy to the study of error correcting codes in theoretical computer science. Alon related the maximum density of a linear graph code which avoids images of a small graph $H$ to the following variant of the Erd\H{o}s-Gy\'arf\'as problem on edge-colourings of $K_n$. A copy of $H$ in an edge-colouring of $K_n$ is even-chromatic if each colour occupies an even number of edges in the copy. We seek an edge-colouring of $K_n$ using $n^{o(1)}$ colours such that there are no even-chromatic copies of $H$. Such an edge-colouring is conjectured to exist for all cliques $K_t$ with an even number of edges. To date, edge-colourings satisfying this property have been constructed for $K_4$ and $K_5$. We construct an edge-colouring using $n^{o(1)}$ colours which avoids even-chromatic copies of $K_8$. This was the smallest open case of the above conjecture, as $K_6, K_7$ each has an odd number of edges. We also study a stronger condition on edge-colourings, where for each copy of $H$, there is a colour occupying exactly one edge in the copy. We conjecture that an edge-colouring using $n^{o(1)}$ colours and satisfying this stronger requirement exists for all cliques $K_t$ regardless of the parity of the number of its edges. We construct edge-colourings satisfying this stronger property for $K_4$ and $K_5$. These constructions also improve upon the number of colours needed for the original problem of avoiding even-chromatic copies of $K_4$ and $K_5$.

math.CO↗

arXiv2024

The probability that a random graph is even-decomposable

A graph $G$ with an even number of edges is called even-decomposable if there is a sequence $V(G)=V_0\supset V_1\supset \dots \supset V_k=\emptyset$ such that for each $i$, $G[V_i]$ has an even number of edges and $V_i\setminus~V_{i+1}$ is an independent set in $G$. The study of this property was initiated recently by Versteegen, motivated by connections to a Ramsey-type problem and questions about graph codes posed by Alon. Resolving a conjecture of Versteegen, we prove that all but an $e^{-\Omega(n^2)}$ proportion of the $n$-vertex graphs with an even number of edges are even-decomposable. Moreover, answering one of his questions, we determine the order of magnitude of the smallest $p=p(n)$ for which the probability that the random graph $G(n,1-p)$ is even-decomposable (conditional on it having an even number of edges) is at least $1/2$. We also study the following closely related property. A graph is called even-degenerate if there is an ordering $v_1,v_2,\dots,v_n$ of its vertices such that each $v_i$ has an even number of neighbours in the set $\{v_{i+1},\dots,v_n\}$. We prove that all but an $e^{-\Omega(n)}$ proportion of the $n$-vertex graphs with an even number of edges are even-degenerate, which is tight up to the implied constant.

math.CO↗

arXiv2024

General characterisation of Hamiltonians generating velocity-independent forces

Dynamics generated from Hamiltonians enjoy potential pathways to quantisation, but standard Hamiltonians are only capable of generating conservative forces. Classes of Hamiltonians have been proposed in Berry et al. capable of generating non-conservative velocity-independent forces. Such Hamiltonians have been classified in the past, under the strict assumption that they are polynomial in momentum. This assumption is relaxed here to analyticity. In doing so, broader classes of Hamiltonians are discovered. By considering the Hamiltonian as a function of state space without introducing the Lagrangian and constructing a metric-like tensor, we develop strong general constraints on Hamiltonians generating velocity-independent forces and exhibit a surprising dichotomy between classes of such Hamiltonians. These results are applicable to any spatial domain of any dimension admitting well-defined Hamiltonian dynamics. As an example application, we apply these constraints to classify all Hamiltonian velocity-independent forces in two spatial dimensions, as well as all such Hamiltonians which do not generate an isotropic simple harmonic motion. The case of one spatial dimension is also discussed for the sake of completeness.

math-ph↗