Publication

Smooth numbers are orthogonal to nilsequences

2025

Abstract

The aim of this paper is to study distributional properties of integers without large or small prime factors. Define an integer to be [y′, y]-smooth if all of its prime factors belong to the interval [y′, y]. We identify suitable weights gy′,y for the characteristic function of [y′, y]-smooth numbers that allow us to establish strong asymptotic results on their distribution in short arithmetic progressions. Building on these equidistribution properties, we show that (a W-tricked version of) the function gy′,y − 1 is orthogonal to nilsequences. Our results apply in the almost optimal range (log N)K < y ⩽ N of the smoothness parameter y, where K ⩾ 2 is sufficiently large, and to any y′ < min(√y, (log N)c ). As a first application, we establish for any y > N1/√log9N asymptotic results on the frequency with which an arbitrary finite complexity system of shifted linear forms ψj (n)+aj ∈ ℤ[n1, …, ns ], 1 ⩽ j ⩽ r, simultaneously takes [y′, y]-smooth values as the ni vary over integers below N.

Official source

https://infoscience.epfl.ch/entities/publication/fd610d41-1dc6-4edc-af12-2141b6728986

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2025

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/fd610d41-1dc6-4edc-af12-2141b6728986

About this result

Related concepts (32)

Smoothness

In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called differentiability class. At the very minimum, a function could be considered smooth if it is differentiable everywhere (hence continuous). At the other end, it might also possess derivatives of all orders in its domain, in which case it is said to be infinitely differentiable and referred to as a C-infinity function (or function).

Smooth number

In number theory, an n-smooth (or n-friable) number is an integer whose prime factors are all less than or equal to n. For example, a 7-smooth number is a number whose every prime factor is at most 7, so 49 = 72 and 15750 = 2 × 32 × 53 × 7 are both 7-smooth, while 11 and 702 = 2 × 33 × 13 are not 7-smooth. The term seems to have been coined by Leonard Adleman. Smooth numbers are especially important in cryptography, which relies on factorization of integers.

Non-analytic smooth function

In mathematics, smooth functions (also called infinitely differentiable functions) and analytic functions are two very important types of functions. One can easily prove that any analytic function of a real argument is smooth. The converse is not true, as demonstrated with the counterexample below. One of the most important applications of smooth functions with compact support is the construction of so-called mollifiers, which are important in theories of generalized functions, such as Laurent Schwartz's theory of distributions.

Related publications (1)

Smooth numbers are orthogonal to nilsequences

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