Power savings for intersective polynomial differences and prime arguments

1 · The essentials

Start with the whole numbers from 1 to some large number $N$ , and choose a collection of them. Require that no difference between two chosen numbers equals a nonzero value produced by a fixed polynomial at a positive whole number. The main result says that, under specific conditions on the polynomial, the collection has size at most a constant times $N^{1 - c}$ , where $c$ is positive. The polynomial must have whole-number coefficients, degree at least two, and a positive leading coefficient. It must also have this remainder property: for every positive whole number, some whole-number input makes its output divisible by that number. The positive amount $c$ can be chosen using only the polynomial’s degree, although the multiplying constant may depend on the polynomial itself. A companion manuscript gives a related bound when only prime inputs are used, under a stronger remainder condition. The main result has a Lean formalization.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine selecting numbered seats in a row while a rulebook forbids certain gaps between occupied seats. Here the polynomial supplies the forbidden gaps, and the result limits how many seats can be occupied; the image alone does not explain why the polynomial’s remainder conditions matter.

Why it matters

A fixed power saving gives a concrete restriction on the size of a collection that avoids the specified differences. For whole-number inputs, the saving exponent is uniform across all qualifying polynomials of a given degree. The square-difference companion is described as answering the fixed-power question posed by Green and Sawhney.

2 · Going further

An integer polynomial $h$ is intersective if, for every positive integer $q$ , there is an integer $r$ with $h (r) \equiv 0 (mod q)$ . The main problem concerns sets $A \subseteq {1, \dots, N}$ such that no difference of elements of $A$ equals a nonzero value $h (n)$ for $n \geq 1$ . For each degree $k \geq 2$ , the main result gives $c_{k} > 0$ such that $∣ A ∣ = O_{h} (N^{1 - c_{k}})$ whenever $h$ is intersective of degree $k$ and has positive leading coefficient. The implied constant may depend on $h$ , but $c_{k}$ depends only on $k$ .

The prime-input companion strengthens the local hypothesis: for every positive integer $q$ , there must be a root $r$ of $h$ modulo $q$ that is coprime to $q$ . Under this unit-root condition, the manuscript shows that avoiding all nonzero differences $h (p)$ with $p$ prime implies $∣ A ∣ \leq C_{h} N^{1 - c_{h}}$ , where $c_{h} > 0$ and $C_{h} \geq 1$ depend only on $h$ . Unlike the main statement, this statement allows the exponent itself to depend on the polynomial.

The abstracts do not describe the argument for the main intersective-polynomial result or the square-difference result. They do identify a key ingredient for the prime-input argument: a companion zero-free half-plane theorem for Dirichlet $L$ -functions supplies the required estimates for the distribution of primes. The source does not state that theorem’s precise region or hypotheses.

The square-difference manuscript treats the case of avoiding nonzero squares and shows a bound $∣ A ∣ \leq C N^{1 - c}$ with absolute constants $c > 0$ and $C < \infty$ . This is the square case within the broader polynomial-difference theme, and its abstract says it answers the fixed-power question posed by Green and Sawhney. The main result has a Lean formalization; the supplied material does not specify its detailed coverage or establish formalization of every companion claim.

Useful notions

Differences between elements of a finite set
Integer polynomials and their degrees
Remainders and congruences
Prime numbers and coprimality
Bounds with constants and exponents

What was known before / what is new

Before

The source identifies a fixed-power question for square-difference-free sets posed by Green and Sawhney. It does not give earlier quantitative bounds or a broader historical account, so no comparison with previous exponents is available here.

New

The main result supplies a fixed power saving for every qualifying intersective polynomial, with the exponent depending only on degree. The prime-input manuscript shows a fixed power saving under the unit-root condition, with polynomial-dependent constants and exponent. The square-difference manuscript shows an absolute-constant bound and describes it as answering the stated fixed-power question.

Limits and assumptions

The polynomial results require a fixed integer polynomial of degree at least two with positive leading coefficient. The whole-number-input result requires intersectivity, while the prime-input result requires a unit root modulo every positive integer; neither statement covers arbitrary polynomials without these hypotheses. Only nonzero polynomial values are forbidden as differences. The main result has a Lean formalization, but the supplied source does not detail its scope, so formal verification is not asserted here for the companion results or the zero-free half-plane ingredient. No explicit saving exponents or remaining open questions are supplied.

Mini glossary

Intersective polynomial: An integer polynomial with a root modulo every positive integer.
Unit root modulo a positive integer: A root modulo that integer represented by a number coprime to it.
Power saving: A bound of the form $C N^{1 - c}$ with a fixed positive exponent $c$ .
Square-difference-free set: A set in which no difference between elements is a nonzero square.
Implied constant: The multiplying constant left implicit in notation such as $O_{h} (N^{1 - c_{k}})$ , where the subscript permits dependence on $h$ .

Keywords : Polynomial differences · Intersective polynomials · Power savings

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

For every fixed intersective integer polynomial h of degree k ≥ 2 with positive leading coefficient, proves that a subset of ${1, \dots, N}$ avoiding nonzero values $h (1), h (2), \dots$ as differences has size $O_{h} (N^{1 - c_{k}})$ , with $c_{k} > 0$ depending only on degree. Here intersective means having a root modulo every modulus. For prime arguments, a power saving also holds when h has a unit root modulo every modulus, with exponent allowed to depend on h.

Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.

Scope of the Lean formalization (source text)

…

lean/docs/182.md

Manuscripts in this family

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A power saving for intersective polynomial differences with an exponent depending only on the degree

05/10/2026

To be verified

An integer polynomial is intersective if it has a root modulo every positive integer. For each degree k ≥ 2, we prove that there is an exponent $c_{k} > 0$ such that every set $A \subseteq {1, \dots, N}$ whose differences avoid all nonzero values $h (1), h (2), \dots$ , where h is an intersective polynomial of degree k with positive leading coefficient, satisfies $∣ A ∣ = O_{h} (N^{1 - c_{k}})$ . The implied constant may depend on h, but the power-saving exponent depends only on its degree.

PDF GitHub folder

A Power Saving for Polynomial Differences at Prime Arguments

05/10/2026

To be verified

Let h be a fixed integer polynomial of degree at least two with positive leading coefficient, having a unit root modulo every positive integer. We prove that any set $A \subseteq {1, \dots, N}$ whose differences avoid all nonzero values $h (p)$ at primes satisfies $∣ A ∣ \leq C_{h} N^{1 - c_{h}}$ , where $c_{h} > 0$ and $C_{h} \geq 1$ depend only on h. Thus the local unit-root condition gives a fixed power saving even when polynomial arguments are restricted to primes. The proof uses the companion zero-free half-plane theorem for Dirichlet L-functions to obtain the required prime-distribution estimates.

PDF GitHub folder

A power saving for square-difference-free sets

24/09/2026

To be verified

We prove that there are absolute constants c > 0 and C < ∞ such that every set $A \subseteq {1, \dots, N}$ with no nonzero square difference satisfies $∣ A ∣ \leq C N^{1 - c}$ . This answers the fixed-power question posed by Green and Sawhney.

PDF GitHub folder