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Junzhe Mao

Publications and source records attributed to Junzhe Mao.

6 recordsLinked to original sources
arXiv2026

A sharp inverse theorem for the quadratic large sieve

We prove that if $A\subseteq[N]$, $|A|\gg\sqrt N$, and $|A_p|\le p/2+O(1)$ for every prime $p\ll \log N$, then $A$ contains at least $\exp\left(c\sqrt{\log N}/\log\log N\right)$ elements in the image of a single integral quadratic $m\pm x^2$. This significantly improves Hanson's logarithmic lower bound, while requiring the residue restriction only for primes $p\ll\log N$. Our proof uses a new weighted entropy argument inspired by our previous work \href{https://arxiv.org/abs/2606.17487}{arXiv:2606.17487} with Sheffer. A matching construction shows that this exponential scale is optimal even when the quadratic may be chosen arbitrarily in $\mathbb{Z}[x]$. We also show extending the residue restrictions to primes up to $(\log N)^3$ yields the stronger bound $\exp(c\sqrt{\log N/\log\log N})$ via a Selberg-sieve-type argument, and discuss an application to the inverse Goldbach problem.

math.NT↗
arXiv2026

The Prouhet--Tarry--Escott problem for subsets with small doubling in integral domains

The Prouhet--Tarry--Escott (PTE) problem has many generalizations and has been studied in various algebraic domains. In this paper, we prove that finite subsets $S$ of integral domains with small additive doubling constant (but still a power of $|S|$) always contain solutions to Wright's generalization of the PTE problem: there are small subsets $A$ and $B$ of the same size such that $\sum_{a\in A} a^j=\sum_{b\in B} b^j$ for $1\le j\le k$, but not for $j=k+1$. More generally, our method gives simultaneous solutions for $m$ systems, with pairwise distinct $(k+1)$-th power sums. In contrast with the classical case $S\subseteq [N]$, where the problem has been studied by Wooley and others using Vinogradov's mean value theorem, our approach is based on polynomial identities and additive properties of $S$. We also discuss barriers to extending these results to broader settings.

math.NT↗
arXiv2026

A combinatorial large sieve for Sidon sets, distances, and norm forms

We develop a new combinatorial large sieve method for sets with bounded algebraic multiplicities. The method exploits algebraic splitting modulo many small primes: local congruence branching produces many modular collisions, while global bounded-multiplicity hypotheses force these collisions to be rare. As a first application, we prove that every Sidon subset $A\subset\{1^2,\ldots,N^2\}$ satisfies \[ |A| \le N\exp\left( -c\frac{\log N}{\log\log N} \right) \] for some absolute constant $c>0$. This gives the first super-polylogarithmic saving for a classical problem of Alon and Erdős. As a second application, we establish new upper bounds for two grid-distance problems. We show that the largest subset of $[N]^2$ with no repeated distance has size at most $N\exp\left(-c\log N/\log\log N\right)$, giving the first progress in over thirty years on a problem of Erdős and Guy. The same method also gives a similar saving for subsets of $[N]^2$ with no isosceles triangles, a problem recently popularized by Ellenberg and by the PatternBoost work of Charton, Ellenberg, Wagner, and Williamson. We then develop an entropic version of the method. This gives bounds for $B_2[g]$-sets in the squares and for analogous bounded-multiplicity problems associated with norm forms over arbitrary number fields. More importantly, this new method also allows us to establish the first nontrivial bounds for $B_3[g]$-sets in the cubes and $B_4[g]$-sets in the fourth powers.

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arXiv2026

Hilbert cubes in sets with arithmetic properties

In this paper, we introduce new general frameworks for estimating the maximal dimension of Hilbert cubes contained in finite truncations of arbitrary sets. As applications, we investigate Hilbert cubes in a range of arithmetic sets, including perfect powers, powerful numbers, primes, smooth numbers, and squarefree numbers. Along the way, we substantially sharpen several earlier results of Dietmann-Elshotlz, Erdős-Sárközy-Stewart, Hajdu, and Sárközy, and we obtain bounds that are sharp up to the implied constant in several cases. Additionally, we prove conditional results of independent interest, including an almost sharp uniform upper bound on the number of $k$-th powers in an arithmetic progression for each $k\geq 4$, assuming the ABC conjecture.

math.NT↗
arXiv2025

An inverse theorem on sets with rich additive structure modulo primes

In this paper, we prove several results on the structure of maximal sets $S \subseteq [N]$ such that $S$ mod $p$ is contained in a short arithmetic progression, or the union of short progressions, where $p$ ranges over a subset of primes in an interval $[y,2y]$ with $(\log N)^{O(1)} < y \leq N$. We also provide several constructions demonstrating the sharpness of our results. Furthermore, as an application, we provide several improvements on the larger sieve bound for $|S|$ when $S$ mod $p$ has strong additive structure, parallel to the work of Green--Harper and Shao for improvements on the large sieve.

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