Algebraic numbers and Fourier analysis: Salem's third problem

Khoa D. Nguyen

Home/Archive/arXiv

arXiv · 2604.18875

Algebraic numbers and Fourier analysis: Salem's third problem

Khoa D. Nguyen

View original source ↗Read source document ↗Export metadata ↓

Abstract

In 1963, Raphaël Salem concluded his highly influential book ``Algebraic Numbers and Fourier Analysis'' with a list of four unsolved problems. The first two problems remain wide open while the last problem on the absolute continuity of Bernoulli convolutions has seen significant progress over the years including recent results by Shmerkin and Varjú. In this paper, we solve the third problem concerning the vanishing at infinity of the product of Fourier transforms of Bernoulli convolutions each of which does not vanish at infinity. Our solution uses tools in diophantine approximation such as the theory of Weil heights and Lang's general formulation of Roth's theorem.

Explore related subjects

math.NT

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Khoa D. Nguyen. 2026-04-20. Algebraic numbers and Fourier analysis: Salem's third problem. https://arxiv.org/abs/2604.18875

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The degrees of irreducible factors of binomials and multiplicativity of the n-Hartley condition

We prove that the degrees of irreducible factors of a binomial $t^n-α$ are divisible by the least such degree. As a consequence, we deduce that for relatively prime natural numbers $m,n$, a polynomial has the $mn$-Hartley condition if and only if it has the $m$-Hartley and $n$-Hartley conditions.

math.NT↗

arXiv2026

On the factorisation of the $p$-adic Rankin-Selberg $L$-function in the supersingular case

Given a cusp form $f$ which is supersingular at a fixed prime $p$ away from the level, and a Coleman family $F$ through one of its $p$-stabilisations, we construct a $2$-variable meromorphic $p$-adic $L$-function for the symmetric square of $F$. We prove that this new $p$-adic $L$-function interpolates values of complex imprimitive symmetric square $L$-functions, for the various specialisations of the family $F$. We use this $p$-adic $L$-function to prove a $p$-adic factorisation formula, expressing the geometric $p$-adic $L$-function attached to the Rankin--Selberg convolution of $f$ with itself as a the product of the $p$-adic symmetric square $L$-function of $f$ and a Kubota-Leopoldt $L$-function. This extends a result of Dasgupta in the ordinary case.

math.NT↗

arXiv2026

Exceptional poles of archimedean Rankin-Selberg L-functions for irreducible generic representations of GL(n,R)

For irreducible generic representations $π_1$ and $π_2$ of $\operatorname{GL}_n(\mathbb R)$, we prove that the notions of exceptional pole of type $1$ and type $2$ coincide at every level. When both representations are in general position, we use this identification to express the Rankin--Selberg $L$-function $L(s,π_1\timesπ_2)$ in terms of the exceptional $L$-factors attached to the irreducible constituents of their derivatives.

math.NT↗