Theorem 1

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Source. Section 4, printed p. 61 (PDF p. 3).

Definitions. Let $G$ be an additive abelian group. A collection $\Omega$ of subsets of $G$ is required to satisfy:

1. if $S_1,S_2\in\Omega$, then $S_1\cup S_2\in\Omega$;
2. if $S_1\in\Omega$ and $S_2\subseteq S_1$, then $S_2\in\Omega$;
3. $G\notin\Omega$;
4. if $x\in G$ and $S\in\Omega$, then $x+S,x-S\in\Omega$.

Members of $\Omega$ are called thin. A set $L\subseteq G\times G$ is called light if there is a thin set $A\subseteq G$ such that the vertical section

$$L_x=\{y\in G:(x,y)\in L\}$$

is thin for every $x\notin A$.

Statement. Let $H$ be an additive abelian group and let $f:G\to H$. Suppose

$$f(x+y)=f(x)+f(y)$$

for every pair $(x,y)$ outside a light subset of $G\times G$. Then there is a homomorphism $h:G\to H$ such that $f(x)=h(x)$ outside a thin subset of $G$.

Proof sketch. The paper says that the proof of Section 2 applies almost literally. A light exceptional set supplies a thin set $M$ of first coordinates whose remaining vertical sections are thin. For fixed $x$, the set $M\cup(x-M)$ is thin and hence is not all of $G$. Choosing $x_1$ outside it makes the two required vertical identities hold outside a thin set of second coordinates. Their sum shows that

$$f(x+y)-f(y)=h(x)$$

outside a thin set depending on $x$. The constant is unique because the union of two thin exceptional sets is not all of $G$. For $x\notin M$, comparison with the original equation gives $h(x)=f(x)$.

The last step repeats the five-exception argument from Section 2. The paper remarks, without written proofs, that coordinate cylinders over thin sets are light, that $\{(w,z):w+z\in S\}$ is light when $S$ is thin, and that translates of light sets are light. A finite union of light sets cannot be all of $G\times G$, so one pair $(w,z)$ satisfies the same five identities used in the real case. Their cancellation proves $h(a+b)=h(a)+h(b)$.

Proof coverage. De Bruijn gives the definitions and the closure facts, but refers to Section 2 for the line-by-line argument. This page records that reduction and the resulting proof structure; a fully expanded group-level version remains to be written.

Dependencies. [[analysis/debruijn_1966_almost_additive_functions/main_theorem|The proof pattern of the main theorem (Section 2)]].

Bears on. #1126