gpt pro 5.2 attempt on Math-656

Erdős Problem #44

Problem statement (from the problem file). Let $N\ge 1$ and $A\subset [N]$ be a Sidon set. Is it true that, for any $\epsilon>0$, there exist $M$ and $B\subset \{N+1,\ldots,M\}$ (which may depend on $N,A,\epsilon$) such that $A\cup B\subset [M]$ is a Sidon set of size at least $(1-\epsilon)M^{1/2}$?

FORMAL RESTATEMENT.

For every $\epsilon>0$, every $N\ge 1$, and every Sidon set $A\subseteq [N]$, does there exist an integer $M\ge N$ and a set $B\subseteq \{N+1,\dots,M\}$ such that $A\cup B$ is Sidon and \[ |A\cup B|\ge (1-\epsilon)\sqrt{M}? \] (Here Sidon is in $\mathbb{Z}$, i.e. $x_1+x_2=x_3+x_4$ with $x_i\in A\cup B$ implies the corresponding multisets of summands are equal.)

QUICK LITERATURE/CONTEXT CHECK.

The problem file indicates links to other problems and mentions Guy's collection; I do not use any unstated external theorems here.

ATTACK PLAN.

Proof track: Try to extend $A$ by adding large elements so that all new sums sit in a range disjoint from the old sums, then iterate carefully to add many elements. Disproof track: Look for a finite Sidon set $A$ that forces "damage" to future densities, e.g. by blocking many residue classes / forcing many forbidden sums. Reality check: Exhaustively verify the statement for very small $N$ and modest search horizon $M$.

WORK.

FAST REALITY CHECK (tiny $N$).

For $N\le 8$, an exhaustive search over all Sidon sets $A\subset [N]$ confirms the following weak finite-horizon version: for $\epsilon=0.2$ and search horizon $M\le 18$, every Sidon $A\subset [N]$ can be extended to some $[M]$ with a Sidon superset of size at least $\lfloor 0.8\sqrt{M}\rfloor$. This is consistent with (but far from evidence for) the asymptotic claim.

Lemma 1 (one-step extension by a huge new element).

Let $S\subset \mathbb{Z}$ be a finite Sidon set and let $m:=\max S$. If $x>2m$ then $S\cup\{x\}$ is a Sidon set.

Proof. Let $T:=S\cup\{x\}$. Suppose $t_1+t_2=t_3+t_4$ with $t_i\in T$. If none of the $t_i$ equals $x$, then the equality is within $S$ and is trivial by Sidon-ness of $S$. If exactly one of the $t_i$ equals $x$, then the sum on that side is $>2m$ while the other side is a sum of two elements of $S$ and is at most $2m$, contradiction. If exactly two of the $t_i$ equal $x$, there are two subcases. If both $x$ occur on the same side then $2x=t_3+t_4\le 2x$ forces $t_3=t_4=x$. If one $x$ occurs on each side, the equality has the form $x+s_1=x+s_2$ with $s_1,s_2\in S$, giving $s_1=s_2$. Thus all solutions are trivial, and $T$ is Sidon. $\square$

Lemma 2 (iterated extension by a rapidly growing tail).

Let $A\subset [N]$ be Sidon and fix an integer $t\ge 1$. Define a sequence by $x_1:=2N+1$ and $x_{i+1}:=2x_i+1$ for $i\ge 1$. Then $A\cup\{x_1,\dots,x_t\}$ is Sidon. In particular, taking $M:=x_t$, one always has an extension to $[M]$ of size $|A|+t$.

Proof. We prove by induction on $i$ that $A_i:=A\cup\{x_1,\dots,x_i\}$ is Sidon. For $i=0$ this holds by hypothesis. Assume $A_i$ is Sidon and let $m_i:=\max A_i = x_i$. By construction $x_{i+1}=2m_i+1>2m_i$, so by Lemma 1, $A_{i+1}=A_i\cup\{x_{i+1}\}$ is Sidon. This completes the induction. $\square$

Remark (why this does not solve the problem).

Lemma 2 yields only $|A\cup B|=|A|+t$ for $M\approx 2^{t}N$, i.e. growth $|A\cup B|=|A|+O(\log M)$, far smaller than the target $(1-\epsilon)\sqrt{M}$. Closing this gap requires adding many elements without pushing $M$ astronomically.

VERIFICATION.

• The "huge element" criterion in Lemma 1 is checked against all mixed cases of a 4-term sum equality.
• Lemma 2 is a direct iteration using the maximal-element condition required by Lemma 1.
• The construction is compatible with the requirement $B\subset\{N+1,\dots,M\}$ because $x_1=2N+1>N$.

FINAL.

UNRESOLVED.

1. (i) Strongest proved partial result. Any Sidon set $A\subset [N]$ can be extended to an infinite Sidon set, and for any prescribed $t$ can be extended by $t$ additional elements at the cost of taking $M$ exponential in $t$ (Lemma 2).
2. (ii) First gap. Show that one can always add linearly many new elements in a range $[N+1,M]$ while preserving Sidon-ness, with $|A\cup B|\ge (1-\epsilon)\sqrt{M}$.
3. (iii) Top 3 next moves. $1$ Develop a quantitative "available slots" counting argument: given current Sidon set $S\subset [M]$ of size $k$, bound how many candidates $x\in [M+1,M+\Delta]$ are forbidden by creating a repeated sum. $2$ Try a block construction: add a translated/affinely transformed near-optimal Sidon set on a new interval and prove cross-sum collisions are avoidable by careful spacing. $3$ Search computationally for small $N$ and worst-case Sidon sets $A\subset [N]$ that maximize the minimal $M$ needed to reach density near $\sqrt{M}$, to guess the correct difficulty and obstructions.
4. (iv) Minimal counterexample structure. A counterexample would be an $\epsilon>0$ and an infinite sequence of pairs $(N,A_N)$ with $A_N\subset [N]$ Sidon such that for every $M\ge N$ and every $B\subset \{N+1,\dots,M\}$ with $A_N\cup B$ Sidon, one has $|A_N\cup B|<(1-\epsilon)\sqrt{M}$.