A uniform common-subsum bound in finite two-groups

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These results use simultaneous representation constraints over every affinely dissociated set of the specified size, rather than one fixed basis. They do not prove the unrestricted odd-set bound or Erdős 963. Affine dissociation means absence of a nonzero balanced ternary relation, with balance taken over the integers, even when the values lie in a finite group.

Binary-tree extraction with variable transversals

Let $G$ be a finite abelian $2$-group. Choose a descending subgroup chain whose successive indices are two, terminating at zero. Its cosets form a binary tree. For any nonempty $X\subseteq G$, the occupied cosets form a finite tree with $|X|$ leaves.

If $|X|>2^{t-2}$, $t\ge2$, there is a path with at least $t-1$ branching vertices. The sibling subtrees at its first $t-1$ branches, and the continuing subtree, partition $X$ into $t$ nonempty clusters. Every transversal is affinely dissociated. Indeed, subtract its point in the continuing cluster from the other selected points. Their differences occupy distinct layers of the subgroup chain. In any nonzero balanced ternary relation, the unique term in the shallowest occupied layer cannot be canceled by deeper-layer terms, as reduction modulo the next subgroup shows.

Consequently

$$|X|\le2^{d_{\rm aff}(X)-1}. \tag{1}$$

There is also a useful rigidity consequence: if every affinely dissociated $t$-subset of $X$ has total zero, then

$$|X|\le\max\{t,2^{t-2}\}. \tag{2}$$

Otherwise the partition above has a nonsingleton cluster. Two transversals differing only in that cluster have distinct totals, contrary to the hypothesis.

Common almost-full subsums

Theorem. Let $r\ge4$ be even and $E\subseteq G$. Suppose there is $k\in G$ such that, for every affinely dissociated $r$-subset $D\subseteq E$,

$$k\in\Sigma_{r-1}(D) :=\left\{\sum_{x\in D\setminus\{a\}}x:a\in D\right\}.$$

Then

$$\boxed{|E|\le2^{r-2}+\max\{r-1,2^{r-3}\}.} \tag{3}$$

The bound is seven for $r=4$, and $3\cdot2^{r-3}$ for every even $r\ge6$.

Multiplication by $r-1$ is invertible on $G$. Translate $E$ by $-(r-1)^{-1}k$; affine dissociation is preserved, and the common subsum becomes zero. If the subgroup $H$ generated by the translated set is zero, the result is immediate. Otherwise choose an index-two subgroup $H_0\subset H$ and put

$$U=E\setminus H_0,\qquad V=E\cap H_0.$$

The generating property gives $U\ne\varnothing$.

No affinely dissociated $r$-set can lie in $U$: each of its $(r-1)$-sums has nonzero image in $H/H_0$, since $r-1$ is odd. Thus $d_{\rm aff}(U)\le r-1$, and (1) gives $|U|\le2^{r-2}$.

For every affinely dissociated $(r-1)$-set $D\subseteq V$ and every $u\in U$, the set $D\cup\{u\}$ is affinely dissociated. A relation using $u$ is excluded modulo $H_0$, and one omitting it is excluded by the hypothesis on $D$. Among this set's $(r-1)$-sums, only $\sum D$ can be zero: every other one uses $u$ exactly once. Hence every such $D$ has total zero. Applying (2) with $t=r-1$ gives $|V|\le\max\{r-1,2^{r-3}\}$, proving (3).

A modular unit-class endpoint

Let $N=2^M$ and let $q=2^s\ge8$ divide $N$. Suppose

$$A\subseteq a+q\mathbb Z/N\mathbb Z,\qquad a\text{ odd}, \qquad d(A)=q,$$

where $d$ is ordinary modular dissociation. Then

$$\boxed{|A|\le2^{q-1}.} \tag{4}$$

The case $q=N$ is vacuous, so split into two classes modulo $2q$. On either fiber, ordinary and affine dimensions agree: any ordinary relation on at most $q+1<2q$ elements has coefficient sum divisible by $2q$, and hence zero. If the ordinary dimension is $d_0\le q$, an affinely dissociated $(d_0+1)$-set would therefore be ordinarily dissociated as well, proving equality of the dimensions.

If both fibers have dimension at most $q-1$, (1) bounds each by $2^{q-2}$, proving (4). Otherwise let $E$ have dimension $q$ and denote the other fiber by $B$. The case $B=\varnothing$ also follows from (1). Multiplication by an odd unit normalizes

$$E\equiv1\pmod{2q},\qquad B\equiv1+q\pmod{2q}.$$

Take any affinely dissociated $q$-set $D\subset E$ and $b\in B$. The set $D$ is ordinarily dissociated. A relation on $D\cup\{b\}$, oriented with coefficient one at $b$, has coefficient sum $s$ on $D$ satisfying $s\equiv q-1\pmod{2q}$ and $|s|\le q$. Thus $s=q-1$. Among $q$ ternary coefficients, this forces exactly $q-1$ plus signs and one zero. Consequently

$$-b\in\Sigma_{q-1}(D) \quad\text{for every such }D.$$

Theorem (3) gives $|E|\le3\cdot2^{q-3}$. Fixing one $D$ also gives $|B|\le q$. Therefore

$$|A|\le3\cdot2^{q-3}+q\le2^{q-1},$$

since $q\le2^{q-3}$ for $q\ge8$.

Remaining limitation

The common-subsum hypothesis in (3) is essential to the proof. For unrestricted odd sets, crossing representations include several mixed-sign coefficient families, not a fixed almost-full subsum family. No reduction to (3) has been established there. In particular (4) does not establish the unrestricted modular odd bound from [[research/erdos_963/finite_group_observations|the cyclic route]], and gives no improvement to the current real-set estimates in the proof sketch.