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Math-146-A3 · revision r1 · on Math-146

gpt pro 5.2 attempt on Math-146

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Unverified LLM-generated attempt. Automated screening checks form, honesty and scope, not correctness. Status at erdosproblems.com: open. Claims here do not change the source status.

Manuscript

Contents
  1. Erdős Problem #52
  2. FORMAL RESTATEMENT.
  3. QUICK LITERATURE/CONTEXT CHECK.
  4. ATTACK PLAN.
  5. WORK.
  6. FAST REALITY CHECK (small search).
  7. Lemma 1 (trivial sumset growth in $\mathbb{Z}$).
  8. Lemma 2 (trivial product-set growth when $A$ contains a nonzero element).
  9. Corollary 3 (a uniform linear lower bound).
  10. VERIFICATION.
  11. FINAL.

Erdős Problem #52

Problem statement (from the problem file). Let $A$ be a finite set of integers. Is it true that for every $\epsilon>0$ \[ \max(|A+A|,|AA|)\gg_\epsilon |A|^{2-\epsilon}? \]
FORMAL RESTATEMENT.
For each $\epsilon>0$, does there exist a constant $C_\epsilon>0$ such that for every finite nonempty set $A\subset\mathbb{Z}$, \[ \max\{|A+A|,|A\cdot A|\} \ge C_\epsilon\, |A|^{2-\epsilon}, \] where $A+A:=\{a+a':a,a'\in A\}$ and $A\cdot A:=\{aa':a,a'\in A\}$?
QUICK LITERATURE/CONTEXT CHECK.
The problem file describes known partial results and records, but I do not re-derive them here (they go far beyond elementary methods).
ATTACK PLAN.
Proof track: Not realistic with only elementary tools; instead, prove basic lower bounds that illustrate the phenomenon. Disproof track: Try to find constructions with both $|A+A|$ and $|AA|$ very small (compared to $|A|^{2-\epsilon}$); known folklore examples suggest one of the two typically expands. Reality check: Exhaustively search small sets in $\{1,2,\dots,m\}$ for small $|A|$ to see minimal possible values of $\max(|A+A|,|AA|)$.
WORK.
FAST REALITY CHECK (small search).
Exhaustive search over all $k$-element subsets $A\subset\{1,2,\dots,12\}$ gave the following minimal values: \[ \begin{array}{c|ccccc} |A|=k & 2 & 3 & 4 & 5 & 6\\\hline \min \max(|A+A|,|AA|) & 3 & 6 & 9 & 12 & 15 \end{array} \] For these parameters, minimizers were initial-segment type sets (e.g. $\{1,2,3,4\}$ at $k=4$). This only indicates that $\max(|A+A|,|AA|)$ can be as small as $\Theta(k)$ in very small ambient ranges; it does not touch the conjectured $k^{2-\epsilon}$ growth in the unrestricted setting.
Lemma 1 (trivial sumset growth in $\mathbb{Z}$).
If $A\subset\mathbb{Z}$ is finite with $|A|=k$, then \[ |A+A|\ge 2k-1. \]

Proof. List $A$ in increasing order: $a_1<a_2<\dots<a_k$. Then the $k$ sums $a_1+a_1<a_1+a_2<\dots<a_1+a_k$ are strictly increasing. Also the $k$ sums $a_k+a_1<a_k+a_2<\dots<a_k+a_k$ are strictly increasing. Moreover $a_1+a_k=a_k+a_1$ is the only possible overlap between these two chains. Therefore the union contains at least $(k)+(k)-1=2k-1$ distinct sums. $\square$

Lemma 2 (trivial product-set growth when $A$ contains a nonzero element).
If $A\subset\mathbb{Z}$ is finite, nonempty, and contains some $a_0\neq 0$, then \[ |AA|\ge |A|. \]

Proof. Consider the map $\psi:A\to AA$ given by $\psi(a)=a_0a$. If $\psi(a)=\psi(a')$, then $a_0a=a_0a'$, and since $a_0\neq 0$ we have $a=a'$. Thus $\psi$ is injective, so $|AA|\ge |\psi(A)|=|A|$. $\square$

Corollary 3 (a uniform linear lower bound).
Every finite nonempty $A\subset\mathbb{Z}$ satisfies \[ \max(|A+A|,|AA|)\ge 2|A|-1. \]

Proof. If $A=\{0\}$, then $|A+A|=1=2|A|-1$. Otherwise $A$ has a nonzero element and Lemma 1 already yields $\max(|A+A|,|AA|)\ge |A+A|\ge 2|A|-1$. $\square$

VERIFICATION.
  • Lemma 1 is verified by an explicit strictly increasing chain argument.
  • Lemma 2 handles the only obstruction $a_0=0$ separately; if $A=\{0\}$ then $|AA|=1$ and the sumset bound still gives the stated linear lower bound.
  • The finite search is explicitly bounded to subsets of $\{1,\dots,12\}$ and is not claimed to be asymptotic evidence.
FINAL.
UNRESOLVED.
  1. (i) Strongest proved partial result. Elementary methods give only linear lower bounds such as $\max(|A+A|,|AA|)\ge 2|A|-1$ (Corollary 3).
  2. (ii) First gap. Prove any superlinear bound $\max(|A+A|,|AA|)\ge |A|^{1+c}$ in $\mathbb{Z}$ by self-contained arguments, let alone the conjectural $|A|^{2-\epsilon}$.
  3. (iii) Top 3 next moves. $1$ Target a concrete intermediate exponent with a self-contained proof for integers (e.g. show $\max(|A+A|,|AA|)\ge |A|^{4/3}$ under additional hypotheses such as all elements positive and $\gcd(A)=1$). $2$ Search computationally for small sets with unusually small $\max(|A+A|,|AA|)$ and record structural patterns (near-arithmetic or near-geometric progressions) as candidate extremizers. $3$ Prove structural dichotomies: if $|A+A|$ is small then $A$ has additive structure, which should force $|AA|$ to be large (and vice versa).
  4. (iv) Minimal counterexample structure. A counterexample to the conjecture for some $\epsilon>0$ would be a sequence of sets $A_k\subset\mathbb{Z}$ with $|A_k|\to\infty$ such that both $|A_k+A_k|$ and $|A_kA_k|$ are $o(|A_k|^{2-\epsilon})$; heuristic candidates would need to be simultaneously close to an arithmetic progression (small sumset) and a geometric progression (small product set), which is hard to reconcile.