we-solve (beta)

Math-2615 · mathematics · Combinatorics

Degree-uniform power saving for intersective polynomial differences

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For each $k>=2$, is there $c_k>0$ such that for every intersective h in Z[x] of degree k with positive leading coefficient, all sufficiently large N and all A subset [N] avoiding nonzero h(t) differences for positive integers t satisfy $|A|<=C_h$ $N^{1-c_k}$, with $C_h$ allowed to depend on h?

Short English statement

For each $k>=2$, is there $c_k>0$ such that for every intersective h in Z[x] of degree k with positive leading coefficient, all sufficiently large N and all A subset [N] avoiding nonzero h(t) differences for positive integers t satisfy $|A|<=C_h$ $N^{1-c_k}$, with $C_h$ allowed to depend on h?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2615-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2615

Degree-uniform intersective differences: The main theorem has one c_k>0 for every k>=2, then for every positive-leading degree-k intersective h constants C_h,N_h, and the exact avoidance (A-A) intersect h(N_+) subset {0}. It allows all signed nonzero polynomial values as differences and an h-dependent threshold; it is not limited to squares or to the exponent depending arbitrarily on h.

With the hypotheses and conventions above, the claimed result addresses the whole mathematical assertion on this page.

Degree-uniform power saving for intersective polynomial differences

OpenAI's claim

Degree-uniform intersective differences: For an intersective polynomial with a positive leading term and degree $k\ge2$, the claimed bound for difference-avoiding subsets is $O_h(N^{1-c_k})$. The exponent depends only on $k$, although the implicit constant depends on the polynomial.

Mathematical statement of the claimed result

Mathematical claim 1.
For each degree $k\ge2$ there is $0<c_k<1$ such that every intersective $h\in\mathbb Z[x]$ of degree $k$ with positive leading coefficient has constants $C_h\ge1,N_h$ satisfying \[(A-A)\cap h(\mathbb N_{>0})\subseteq\{0\},\quad A\subseteq\{1,\ldots,N\},\quad N\ge N_h\quad\Longrightarrow\quad|A|\le C_hN^{1-c_k}.\] Intersective means having a root modulo every positive integer. The saving exponent depends only on the degree; the prefactor and threshold may depend on all coefficients.

Outline of the argument

Degree-uniform intersective differences: Local roots are normalized until their first nonzero Taylor coefficients are units, without changing their multiplicities. Auxiliary polynomials are chosen to remain compatible under passage to progressions. A multilinear density lower bound and a distribution-comparison argument then produce contraction with numerical choices controlled by the degree rather than the coefficients.

Formal verification

Family 182 describes Lean coverage for the square-difference power saving; the general polynomial and prime-input estimates are separate.

Degree-uniform intersective differences: No manuscript-specific Comparator is listed in the pinned inventory.

The documented selection does not cover the entire manuscript assertion relevant here.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources

Go to contribution →