arXiv subjects

Dijia Chen

Publications and source records attributed to Dijia Chen.

5 recordsLinked to original sources

arXiv2026

A Counterexample to Wooley's $L^1$ Question for Beatty Sequences

For real numbers $α$ and $β$, let $\mathcal{B}(α,β)=\{\lfloorαm+β\rfloor:m\in\mathbb{N}\}$. Wooley asked whether the exponential sum over $\mathcal{B}(α,β)\cap[1,N]$ always has $L^1$ norm $O_{α,β}(\log N)$. We give a negative answer. We construct fixed $α>1$ and $β$, with $1/α$ of finite Diophantine type, such that the $L^1$ norm is $\gg(\log N)^2$ for infinitely many $N$. Wooley also proved that this $L^1$ norm is $O_{α,β}((\log(2N))^2)$ when $1/α$ has finite Diophantine type. Our example shows that his upper bound is sharp.

math.NT↗

arXiv2026

Rigidity of Averages over the Two Largest Prime Factors

Let $P_1(n)$ and $P_2(n)$ be the largest and second-largest distinct prime factors of $n$. Alladi and Johnson asked whether there is a bounded function $f$ on the primes such that the averages $\frac{1}{x}\sum_{2\le n\le x}f(P_1(n))$ and $\frac{1}{x}\sum_{2\le n\le x}f(P_2(n))$ converge to different limits, where $f(P_2(n))=0$ when $n$ is a prime power. We show that this cannot happen. In fact, whenever the first average converges, the second also converges to the same limit.

math.NT↗

arXiv2026

Hecke Eigenform Product Identities for Full-Level Hilbert Modular Forms

We study full-level Hilbert Hecke eigenform product identities over totally real fields of degree at least two, allowing arbitrary compatible narrow class characters. Combining our results with previous work, we classify products of two Eisenstein eigenforms of positive parallel integral weights and products of an Eisenstein eigenform of weight at least two with a cuspidal eigenform of positive parallel integral weight. The unique identity in each of these two classes is given by $E_{4}=60 E_{2}^{2}$ and $h_{8}=120 E_{2} \cdot h_{6}$, respectively, both over $\mathbb{Q}(\sqrt{5})$. The proofs use constant-term comparisons, indecomposable Fourier exponents, integrality of reciprocal Eisenstein constant terms, and a Rankin-Selberg formula for Petersson inner products. In the remaining case of a weight-one Eisenstein factor and a cuspidal factor, we also establish $h_5=12E_1(1,\varepsilon)\cdot h_4$ over $\mathbb Q(\sqrt3)$, where $\varepsilon$ is the nontrivial narrow class character. All eigenforms in these identities have unit-ideal coefficient one.

math.NT↗

arXiv2026

Bounded Ratios of Lorentzian Polynomials I: The Ternary Theory and Optimal Bounding Constants

We study bounded ratios and optimal bounding constants among the normalized coefficients of ternary Lorentzian polynomials. For every fixed $M$-convex support and in arbitrary degree, we give an explicit presentation of the bounded-ratio cone in terms of quadratic Hessian slices. We then express the optimal bounding constants through a variational formula combining local support functions with linear compatibility constraints between slices. For full support, we determine all compatibility relations in arbitrary degree; in degree three, this yields explicit optimal constants for every two-generator section. Finally, we compare the resulting Lorentzian bounds with those for volume polynomials and rank-three matroid basis profiles.

math.CO↗

arXiv2026

Bounded Ratios of Lorentzian Polynomials II: The Complete Quadratic Local-to-Global Classification

Every quadratic Hessian slice of a Lorentzian polynomial yields bounded monomial ratios among the normalized coefficients of the polynomial. We determine exactly for which pairs $(n,d)$ these quadratic-slice ratios generate the full bounded-ratio cone for every $M$-convex support $S\subseteqΔ_n^d$. For $d\geq 2$, this quadratic local-to-global principle holds universally if and only if \[ n\leq 3,\qquad d=2,\qquad\text{or}\qquad (n,d)=(4,3). \] In every remaining case, the principle fails already for Lorentzian polynomials with full support: for cubics in $n\geq 5$ variables and for polynomials of degree $d\geq 4$ in $n\geq 4$ variables. We identify the two minimal obstructions, at $(n,d)=(4,4)$ and $(n,d)=(5,3)$, and propagate them throughout the failure region by degree and variable aggregation.

math.CO↗