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Mehtaab Sawhney

Publications and source records attributed to Mehtaab Sawhney.

At least 19 recordsLinked to original sources
arXiv2026

Primes of the form $p^2 + nq^2$

Suppose that $n$ is $0$ or $4$ modulo $6$. We show that there are infinitely many primes of the form $p^2 + nq^2$ with both $p$ and $q$ prime, and obtain an asymptotic for their number. In particular, when $n = 4$ we verify the `Gaussian primes conjecture' of Friedlander and Iwaniec. We study the problem using the method of Type I/II sums in the number field $\mathbf{Q}(\sqrt{-n})$. The main innovation is in the treatment of the Type II sums, where we make heavy use of two recent developments in the theory of Gowers norms in additive combinatorics: quantitative versions of so-called concatenation theorems, due to Kuca and to Kuca--Kravitz-Leng, and the quasipolynomial inverse theorem of Leng, Sah and the second author.

math.NT↗
arXiv2026

The proportion of permutations fixing a $k$-set

Denote by $p(k)$ the limit, as $n \rightarrow \infty$, of the probability that a random permutation on a set of size $n$ has an invariant set of size $k$. We give an asymptotic formula for $p(k)$, showing that it is asymptotically $f(\{\log_2 k\}) k^{-δ} (\log k)^{-3/2}$ where $δ= 1 - \frac{1 + \log \log 2}{\log 2} \approx 0.086$ and $f$ is a smooth, positive, function on $\mathbb{R}/\mathbb{Z}$, which we will describe explicitly. The function $f$ satisfies $\frac{\max f}{\min f} < 1 + 2 \times 10^{-7}$ and we conjecture that it is not constant. Estimating $p(k)$ is a model for the more well-known question which asks for an estimation of $M(n)$, the number of distinct elements in the $n$-by-$n$ multiplication table. By elaborating on the techniques in this paper, we will give an asymptotic for $M(n)$ in forthcoming work.

math.CO↗
arXiv2026

On the maxima of Littlewood polynomials on $[-1,1]$

A Littlewood polynomial is a polynomial of the form \[ f_n(x)=\sum_{k=0}^n \varepsilon_k x^k \] with $\varepsilon_k\in\{-1, 1\}$. Let $(\varepsilon_k)_{k \ge 0}$ be i.i.d. Rademacher coefficients. We show that the lower envelope of $\max_{x\in[-1,1]}|f_n(x)|$ is determined by the small-ball probability of a certain Gaussian process. In particular, almost surely, \[ \liminf_{n\to\infty} \frac{\log(\max_{x\in[-1,1]}|f_n(x)|/\sqrt n)}{(\log\log n)^{1/3}} = -\Big(\frac{3π^2}{4}\Big)^{1/3}. \]

math.PR↗
arXiv2026

Short proofs in combinatorics, probability and number theory II

We give a quintet of proofs resulting from questions posed by Erdős. These questions concern ordinary lines in planar point sets, sequences with uniformly small exponential sums, $K_4$-free $4$-critical graphs with few chords in any cycle, a counterexample to a "fewnomial" version of the Erdős--Turán discrepancy bound, and a finiteness theorem for integers $n$ such that $n-a k^2$ is prime for all $k\leq \sqrt{n/a}$ coprime to $n$ (for fixed $a\in\mathbb Z_+$). Each proof is due to an internal model at OpenAI.

math.CO↗
arXiv2026

Short proofs in combinatorics and number theory

We give a triplet of short proofs, each of which answers a question raised by Erdős. The first concerns the small prime factors of $\binom{n}{k}$, the second concerns whether an additive basis $A$ can always be split into pieces $A_1$ and $A_2$ such that each of $A_i + A_i$ has bounded gaps, and the final concerns whether $\{αp\}$ is "well-distributed" in the sense introduced by Hlawka and Petersen. In each case, the proof is due entirely to an internal model at OpenAI.

math.CO↗
arXiv2026

On infinite sets with no $3$ on a line

We give a construction of an infinite set of points $A$ in $\mathbb{R}^2$ such that any subset $P\subseteq A$ has a constant density subset $P'$ with no three points collinear and yet $A$ cannot be separated into finitely many subsets such that each subset has no three points collinear. This provides a new proof of a question of Erdős, Nešetřil, and Rödl. The construction was generated by an internal model at OpenAI.

math.CO↗
arXiv2025

Early science acceleration experiments with GPT-5

AI models like GPT-5 are an increasingly valuable tool for scientists, but many remain unaware of the capabilities of frontier AI. We present a collection of short case studies in which GPT-5 produced new, concrete steps in ongoing research across mathematics, physics, astronomy, computer science, biology, and materials science. In these examples, the authors highlight how AI accelerated their work, and where it fell short; where expert time was saved, and where human input was still key. We document the interactions of the human authors with GPT-5, as guiding examples of fruitful collaboration with AI. Of note, this paper includes four new results in mathematics (carefully verified by the human authors), underscoring how GPT-5 can help human mathematicians settle previously unsolved problems. These contributions are modest in scope but profound in implication, given the rate at which frontier AI is progressing.

cs.CL↗
arXiv2025

On random matrices with large corank

Let $1\le k\le n$ and $M$ be a random $n\times n$ matrix with independent uniformly random $\{\pm 1\}$-entries. We show that there exists an absolute constant $c > 0$ such that \[\mathbf{P}[\operatorname{rank}(M)\le n-k]\le \exp(-c nk).\]

math.PR↗
arXiv2025

Parities in random Latin squares

In a Latin square, every row can be interpreted as a permutation, and therefore has a parity (even or odd). We prove that in a uniformly random $n\times n$ Latin square, the $n$ row parities are very well approximated by a sequence of $n$ independent unbiased coin flips: for example, the total variation error of this approximation tends to zero as $n\to\infty$. This resolves a conjecture of Cameron. In fact, we prove a generalisation of Cameron's conjecture for the joint distribution of the row parities, column parities and symbol parities (the latter are defined by the symmetry between rows, columns and symbols of a Latin square). Along the way, we introduce several general techniques for the study of random Latin squares, including a new re-randomisation technique via `stable intercalate switchings', and a new approximation theorem comparing random Latin squares with a certain independent model.

math.PR↗
arXiv2025

Convergent points for random power series on the unit circle

Consider a random power series of the form $P(z) = \sum_{n\ge 1} \varepsilon_n a_n z^{n}$ where $a_n \in \mathbb{C}$ are deterministic and $\varepsilon_n$ are chosen independently and uniformly at random from $\{\pm 1\}$. Kolmogorov's three-series theorem states that if $\sum_{n} |a_n|^2 = \infty$ then $P(z)$ almost-surely diverges at almost every $z$ with $|z| = 1$. Dvoretzky and Erdős proved in 1959 that if $|a_n| = Ω(1/\sqrt{n})$ then in fact $P$ almost surely diverges at every $|z| = 1$. Erdős then asked in 1961 if this is sharp, meaning that if $|a_n| = o(1/\sqrt{n})$ then there is almost surely some convergent point $z$ with $|z| = 1$. We prove this in a strong sense and show that if $a_n = o(1/\sqrt{n})$ then in fact the set of convergent points of $P$ with $|z| = 1$ has Hausdorff dimension $1$.

math.PR↗
arXiv2025

Quasipolynomial bounds for the corners theorem

Let $G$ be a finite abelian group and $A$ be a subset of $G \times G$ which is corner--free, meaning that there are no $x, y \in G$ and $d \in G \setminus \{0\}$ such that $(x, y)$, $(x+d, y)$, $(x, y+d) \in A$. We prove that \[|A| \le |G|^2 \cdot \exp(-(\log |G|)^{Ω(1)}).\] As a consequence, we obtain polynomial (in the input length) lower bounds on the nondeterministic communication complexity of Exactly-N in the 3-player Number-on-Forehead model. We also obtain the first "reasonable'' lower bounds on the coloring version of the $3$-dimensional corners problem, as well as on the nondeterministic communication complexity of Exactly-N in the 4-player Number-on-Forehead model.

math.CO↗
arXiv2025

The limiting spectral law for sparse iid matrices

Let $A$ be an $n\times n$ matrix with iid entries where $A_{ij} \sim \mathrm{Ber}(p)$ is a Bernoulli random variable with parameter $p = d/n$. We show that the empirical measure of the eigenvalues converges, in probability, to a deterministic distribution as $n \rightarrow \infty$. This essentially resolves a long line of work to determine the spectral laws of iid matrices and is the first known example for non-Hermitian random matrices at this level of sparsity.

math.PR↗
arXiv2025

On polynomial progressions via transference

We prove new cases of reasonable bounds for the polynomial Szemerédi theorem both over $\mathbb{Z}/N\mathbb{Z}$ with $N$ prime and over the integers. In particular, we prove reasonable bounds for Szemerédi's theorem in the integers with fixed polynomial common difference. That is, we prove for any polynomial $P(y)\in \mathbb{Z}[y]$ with $P(0) = 0$, that the largest subset $A\subseteq [N]$ avoiding the pattern \[x, x+P(y),\ldots, x+ kP(y)\] has size bounded by $\ll_{P,k}N(\log\log\log N)^{-Ω_{P,k}(1)}.$

math.NT↗
arXiv2025

A central limit theorem for the matching number of a sparse random graph

In 1981, Karp and Sipser proved a law of large numbers for the matching number of a sparse Erdős-Rényi random graph, in an influential paper pioneering the so-called differential equation method for analysis of random graph processes. Strengthening this classical result, and answering a question of Aronson, Frieze and Pittel, we prove a central limit theorem in the same setting: the fluctuations in the matching number of a sparse random graph are asymptotically Gaussian. Our new contribution is to prove this central limit theorem in the subcritical and critical regimes, according to a celebrated algorithmic phase transition first observed by Karp and Sipser. Indeed, in the supercritical regime, a central limit theorem has recently been proved in the PhD thesis of Kreačić, using a stochastic generalisation of the differential equation method (comparing the so-called Karp-Sipser process to a system of stochastic differential equations). Our proof builds on these methods, and introduces new techniques to handle certain degeneracies present in the subcritical and critical cases. Curiously, our new techniques lead to a non-constructive result: we are able to characterise the fluctuations of the matching number around its mean, despite these fluctuations being much smaller than the error terms in our best estimates of the mean. We also prove a central limit theorem for the rank of the adjacency matrix of a sparse random graph.

math.CO↗
arXiv2025

The random graph process is globally synchronizing

The homogeneous Kuramoto model on a graph $G = (V,E)$ is a network of $|V|$ identical oscillators, one at each vertex, where every oscillator is coupled bidirectionally (with unit strength) to its neighbors in the graph. A graph $G$ is said to be globally synchronizing if, for almost every initial condition, the homogeneous Kuramoto model converges to the all-in-phase synchronous state. Confirming a conjecture of Abdalla, Bandeira, Kassabov, Souza, Strogatz, and Townsend, we show that with high probability, the random graph process becomes globally synchronizing as soon as it is connected. This is best possible, since connectivity is a necessary condition for global synchronization.

math.CO↗
arXiv2025

The sparse circular law, revisited

Let $A_n$ be an $n\times n$ matrix with iid entries distributed as Bernoulli random variables with parameter $p = p_n$. Rudelson and Tikhomirov, in a beautiful and celebrated paper, show that the distribution of eigenvalues of $A_n \cdot (pn)^{-1/2}$ is approximately uniform on the unit disk as $n\rightarrow \infty$ as long as $pn \rightarrow \infty$, which is the natural necessary condition. In this paper we give a much simpler proof of this result, in its full generality, using a perspective we developed in our recent proof of the existence of the limiting spectral law when $pn$ is bounded. One feature of our proof is that it avoids the use of $ε$-nets entirely and, instead, proceeds by studying the evolution of the singular values of the shifted matrices $A_n-zI$ as we incrementally expose the randomness in the matrix.

math.PR↗