gpt 6 astra pro attempt on Math-146

Erdős Problem #52

Definitions and mathematical statement

For a finite set $A\subset{\mathbb{Z}}$, write $A+A=\{a+b:a,b\in A\}$ and $A\cdot A=\{ab:a,b\in A\}$. The conjecture is \[ \forall\varepsilon>0\ \exists c_\varepsilon>0\ \forall A\subset{\mathbb{Z}}\text{ finite}:\qquad \max(|A+A|,|A\cdot A|)\ge c_\varepsilon|A|^{2-\varepsilon}. \] The constant is uniform in the elements and cardinality of $A$. The assertion says that addition and multiplication cannot both produce exceptionally small image sets. It does not require both sets to have nearly quadratic size.

Formulation and concept references: [S1].

Short English statement

Must a finite integer set expand almost quadratically under either addition or multiplication?

Research attempt

Current frontier and source audit.

The original source's introduction states the integer conjecture, while most of that paper investigates sparse-graph variants. We retain the full sum and product sets here. The real-number version is not interchangeable with this target. [S1]

Cushman's December 2025 manuscript, Theorem 1.3, gives the general-real exponent $4/3+10/4407-\epsilon$, hence also a lower bound for integer sets. Bloom--Sawin--Schildkraut--Zhelezov's May 2026 manuscript, Theorem 1.1, constructs counterexamples over real number fields whose degrees grow with the set size; its introduction explicitly leaves the integer case separate. These are identified as research manuscripts, not independently verified by this notebook. No part of either proof is needed for the lemmas below. [E1][E2]

Attack route.

A purely real-geometric route must now distinguish itself from the real counterexample construction. We instead combine exact integer divisibility with an analysis of how a large translation affects product collisions. A first height-dependent divisor bound becomes a bound uniform in both the translation and the step of a containing arithmetic progression. The remaining issue is to replace that one-dimensional progression hypothesis by consequences of small additive doubling without destroying multiplicative relations.

Attempt: the sharp translated-interval threshold.

For a finite positive set $A$, call its products injective on unordered pairs if $ab=cd$ with $a,b,c,d\in A$ implies $\{a,b\}=\{c,d\}$ as multisets; repetitions such as $a=b$ are allowed.

Lemma 1. For integers $D\geq1$ and $M>0$, every subset of $[M,M+D]\cap{\mathbb{Z}}$ has injective unordered products whenever \begin{equation}\label{wsref-0-65-71-3a-31-39-30-34-3a-74-68-72-65-73-68-6f-6c-64} M>\left\lfloor\frac{(D-1)^2}{4}\right\rfloor. \end{equation} For every $D\geq3$ the integer threshold is sharp for the full interval.

Proof. More generally consider the real translate $t+B$, where $t>0$ and $B\subseteq\{0,\ldots,D\}$. A collision implies \[ t(x+y-z-w)=zw-xy. \] If the sums agree, so do the products, and the roots of the corresponding quadratic show that the unordered pairs agree. Otherwise exchange the pairs so that $x+y=s$, $z+w=s+\delta$, with integer $\delta\geq1$. Then $t\delta=xy-zw>0$. The elementary bounds \[ xy\leq\frac{s^2}{4},\qquad zw\geq\max\{0,D(s+\delta-D)\} \] show that no such positive collision is possible when $\delta>D$. For $1\leq\delta\leq D$, the difference of these bounds is largest at $s=D-\delta$: below that value it is increasing, and above it its derivative is $s/2-D\leq0$. Consequently \[ t\leq\frac{(D-\delta)^2}{4\delta} \leq\frac{(D-1)^2}{4}. \] For integer $t=M$ this proves (eq:1904:threshold). If $D=2k+1$, the collision at $M=k^2$ is \[ (M+k)^2=M(M+2k+1). \] If $D=2k$, the collision at $M=k(k-1)$ is \[ (M+k-1)(M+k)=M(M+2k). \] Both have different unordered pairs, proving sharpness. $\square$

The lemma gives $|AA|=|A|(|A|+1)/2$ in that regime. It also shows why large absolute height, by itself, is not a plausible source of counterexamples: a sufficiently remote short interval has maximal unordered product growth.

Attempt: eliminate the step size by a gcd dichotomy.

Lemma 2. Fix $C>0$ and $\epsilon>0$. There is a constant $c_{C,\epsilon}>0$ such that every $n$-element integer set contained in \[ a+g\{0,1,\ldots,D\},\qquad g\in{\mathbb{Z}}\setminus\{0\},\quad 0\leq D\leq n^C, \] satisfies $|AA|\geq c_{C,\epsilon}n^{2-\epsilon}$. The constant is independent of $a$, $g$, and the elements of $A$.

Proof. Discard zero and take either the positive or the negative part with at least $(n-1)/2$ elements. Negating the latter does not change its product cardinality. This produces a positive subset $A_0$, with $m\geq(n-1)/2$, which can be written \[ A_0\subseteq M+g\{0,\ldots,D\},\qquad M>0,\quad g>0, \] after shrinking and reindexing the containing progression if necessary. The finitely small cardinalities can be absorbed into the constant. Divide all elements by $d=\gcd(M,g)$, putting $M'=M/d$, $g'=g/d$; this preserves product cardinality and gives $\gcd(M',g')=1$.

A product collision among $M'+g'x$ and its companions gives \begin{equation}\label{wsref-0-65-71-3a-31-39-30-34-3a-64-69-76-69-73-69-62-69-6c-69-74-79} M'(x+y-z-w)=g'(zw-xy). \end{equation} If $g'>2D$, divisibility forces $x+y-z-w=0$, because its absolute value is at most $2D$; then the unordered pairs are equal. If $M'/g'>(D-1)^2/4$, the real-translation version of Lemma 1 also gives injectivity. In either case $|A_0A_0|=m(m+1)/2$.

In the only remaining case, \[ g'\leq2D,\qquad M'\leq g'(D-1)^2/4, \] all normalized elements are positive integers at most $3D^3$ for $D\geq1$. Their products are at most $9D^6$. For completeness, for every $\delta>0$ the divisor function satisfies \begin{equation}\label{wsref-0-65-71-3a-31-39-30-34-3a-64-69-76-69-73-6f-72} d(v)\leq C_\delta v^\delta\qquad(v\geq1). \end{equation} Indeed, in $v=\prod p^{e_p}$, for sufficiently large primes $e+1\leq2^e\leq p^{\delta e}$; for each of the finitely many smaller primes the supremum of $(e+1)p^{-\delta e}$ is finite. Multiplying these bounds proves (eq:1904:divisor).

For each positive product $v$, at most $d(v)$ ordered pairs of positive integers have that product. Thus \[ m^2\leq |A_0A_0|\,C_\delta(9D^6)^\delta \leq |A_0A_0|\,C_\delta9^\delta n^{6C\delta}. \] Taking $\delta=\epsilon/(6C)$ and using $|AA|\geq|A_0A_0|$ proves the lemma. The case $D=0$ has at most one element and is harmless. $\square$

This proves a substantial restricted form without using $|A+A|$ at all. It includes sets of polynomial diameter, but also progressions with arbitrarily large step and translation. The argument is supplied as a self-contained deduction, without asserting that the result is new in the literature.

Attempt to remove the progression hypothesis.

Small additive doubling often suggests modeling a set by a generalized arithmetic progression. But neither an arbitrary additive model nor a translation is a multiplicative homomorphism. The simple divisibility identity (eq:1904:divisibility) involves only one step; with several steps, independent linear and quadratic expressions in those steps enter, and no one-dimensional coprimality argument eliminates all of them. The precise missing statement for this route is a uniform product-growth or sum-growth dichotomy for arbitrary integer sets of large normalized progression span, not a height estimate with an element-dependent constant.

There is also no automatic descent of the 2026 real construction to integers. For example, if $\alpha$ is an irrational algebraic integer with irreducible polynomial $f$, there is no unital ring homomorphism ${\mathbb{Z}}[\alpha]\to{\mathbb{Z}}$: the image of $\alpha$ would be an integer root of $f$. A finite encoding preserving the additions and multiplications expressing $f(\alpha)=0$ and fixing integers has the same obstruction. Substituting a large integer for $\alpha$ keeps some additive information but breaks that exact multiplicative relation. This does not rule out every possible finite combinatorial transfer; it rules out the naive ring-specialization step. The growing number-field degree in the manuscript is therefore an actual target distinction, not just a cosmetic normalization. [E2]

Stress test and exact computations.

The companion script checks the sharp boundary collisions and the absence of collisions above the threshold for $3\leq D\leq30$. It also tests the gcd dichotomy for many progressions, including steps larger than $2D$. These checks test the formulas; the proofs, not the finite search, establish the lemmas for every permitted parameter.

The set $A=\{1,2,4,\ldots,2^{n-1}\}$ has $|AA|=2n-1$, so no universal near-quadratic product-only lemma is possible. Its unordered sums, however, are all distinct by binary expansion, including the doubled terms, and $|A+A|=n(n+1)/2$. It is a test against the proposed intermediate assertion, not against the actual max-of-two-sets conjecture. It also has exponential span and lies outside the hypotheses just proved.

Outcome.

Outcome: Exploratory approach with unresolved gap.

Two exact lemmas were obtained: a sharp product-injectivity threshold for translated integer intervals, and near-quadratic product growth for sets in polynomial-length arithmetic progressions, uniformly in the progression's height and step. The remaining arbitrary-set reduction is unproved. No integer counterexample, full sum-product proof, or improvement to the unrestricted exponent is claimed.

Sources

• S1 Noga Alon, Omer Angel, Itai Benjamini, and Eyal Lubetzky, Sums and products along sparse graphs, Israel Journal of Mathematics 188 $2012$, 353-384. https://www.math.tau.ac.il/~nogaa/PDFS/sumprod2.pdf. Catalogue formulation source.
• E1 Adam Cushman, A Note on the Sum-Product Problem and the Convex Sumset Problem, arXiv:2512.13849v1 (15 December 2025), Theorem 1.3. https://arxiv.org/html/2512.13849v1.
• E2 Thomas F. Bloom, Will Sawin, Carl Schildkraut and Dmitrii Zhelezov, The sum-product conjecture is false for real numbers, arXiv:2605.28781v1 (27 May 2026), Theorem 1.1 and Sections 1--2. The real/algebraic-number target is not the integer target of this entry. https://arxiv.org/html/2605.28781v1.