Constructing small maximal Sidon sets

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consecutive_differences: Sliding interval counts show that m integers cannot represent every difference from 1 to L uniquely unless L is O(m^(5/3)), irrespective of their diameter. Thus a Sidon difference basis at quadratic scale cannot supply a carry-free construction for Problem 156.

evidence/: Check finite carry identities, anchored-lift residuals, a coupled completion witness, two- and three-translate block coverings, the retained consecutive-difference rulers, and multiple-height seeds and completions.

finite_templates: A retained 40-point Sidon seed with 399 misses completes with eight additions in [1,8001]. Exact censuses test two and three translates of fixed Singer difference sets as local covering templates. Neither finite result establishes a uniform construction.

integer_transfer: Why the exact group constructions do not transfer to integer intervals: in an anchored Singer lift the main-type blocking triples have pointers shifted by carries. Equal residues give one coherence mechanism, while long intervals of uniquely represented differences face a separate bound.

multiple_heights: Disjoint internal height differences characterize compatible fibers above a modular Sidon set. Residuals confined to few classes or rows admit small completions. Exact finite examples measure the gap between these sufficient criteria and the coverage of explicit height systems.

proof_sketch: Editable sketch of the attack on Problem 156: exact group constructions at the cube-root scale are held; the transfer to integer intervals is the open gap, with each route tried and its obstruction.

translated_difference_sets: If a Sidon set of k integers has a translate s+(W-W) inside its difference set with |s| larger than the diameter of W, then |W| <= (3/2+sqrt 2)k^(5/3). Consequences bound progressions, combs, translated Bohr parameters and linear sign functions, and exclude covering an interval through one separated piece of bounded doubling at the cube-root scale.

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This folder records an attack on Problem 156: for every large $N$, find a maximal strong Sidon set in $\{1,\dots,N\}$ of size $O(N^{1/3})$, or show that none exists. The conventions, the counting lower bound and the Ruzsa benchmark are those of the research guide for Problem 156. The problem remains open.

The attack has settled the question in two families of finite groups, by explicit constructions in which a mixed blocking triple has exactly one solution for almost every target, so that coverage needs no probability.

• L208 gives a maximal strong Sidon set of $2q-1$ elements in $\mathbb F_q^3$ for every odd prime power $q$.
• L209 gives maximal strong Sidon sets of size $p+h$ in groups $G=\mathbb Z_q\times H$ with $q=p^2+p+1$ and $|H|=h$ odd, $h\le p/4$. For $h\ge p/8$ this is $O(|G|^{1/3})$, and the group is cyclic for suitable cyclic $H$.

Both are tier 0, author-recorded by Plasma AI.

The proof sketch is the editable account of the attack: target, held links at their standing, the remaining gap and the routes tried. The integer transfer analysis isolates the extra carry condition for integer coverage. The integer analogue of L209 is the anchored Singer lift, and there the residue sum of each blocking triple may carry into the height. It records the exact structure of that carry, one sufficient coherence mechanism, and the conditions under which the route should be reopened.

The sliding-count argument excludes a proposed carry-free alternative: $m$ integers cannot represent every difference from $1$ to a constant times $m^2$ uniquely. Its bound is $O(m^{5/3})$, without a diameter restriction. It does not exclude several translated difference sets covering one another's gaps, or small coupled repairs of a seed with many initial misses.

The bound of [[theory/additive_bases/L272_translated_unique_differences/_index|L272, at author-recorded tier 0]], extends that count. If a Sidon set of $k$ integers has a translate $s+(W-W)$ in its difference set with $|s|$ larger than the diameter of $W$, then $|W|=O(k^{5/3})$. The research consequences bound progressions, combs and translated Bohr parameters, and exclude a Sturmian sign function for perfect difference sets of sufficiently large order. Coverage using one separated difference-set piece of bounded doubling would require $\Omega(N^{3/8})$ points. That restriction does not exclude a general cubic-root construction.

The multiple-height analysis gives an exact fiber compatibility criterion and explicit cubic-scale Sidon seeds. It also proves that residuals confined to $O(p)$ classes modulo $q\asymp p^2$, or to $O(1)$ intervals of $q$ consecutive integers, admit $O(p)$ completion when $N\asymp p^3$. Its finite witnesses test the gap between compatible fibers and this sufficient concentration property. No uniform integer construction or completion theorem is established.