---

Source. Theorem 4, p. 228, of M. Laczkovich, Functions with measurable differences, Acta Mathematica Academiae Scientiarum Hungaricae 35 (1980), 217--235, the edition named on the source card.

Read depth. Claims checked: the statement and the definitions it uses (p. 217 and p. 227) were read clause by clause on the page images; the proof (pp. 228--229) was read for structure only. Nothing here is independently reviewed.

Statement

Setting. The weak difference property of a class $F$ (p. 217) is defined on the Theorem 3 page. $L_p(0,1)$ (p. 227) is the class of Lebesgue measurable $f:\mathbb R\to\mathbb R$ that are periodic mod 1 and satisfy $\|f\|_p=\bigl(\int_0^1|f(x)|^p\,dx\bigr)^{1/p}<\infty$.

Theorem 4 (p. 228, quoted). "The classes $L_p(0,1)$ have the weak difference property for every $p>0$."

Unwound: let $p>0$. If $f:\mathbb R\to\mathbb R$ and $f(x+h)-f(x)\in L_p(0,1)$ for every real $h$, then $f=g+H+S$ with $g\in L_p(0,1)$, $H$ additive, and, for each fixed $h$, $S(x+h)-S(x)=0$ for almost every $x$.

The introduction (p. 217) says the weak difference property was known for $p\ge1$ (the paper's reference [4], with a generalization in [13]) and that F. W. Carroll asked whether it holds for $0<p<1$; Theorem 4 answers that question affirmatively.

Proof pointer

Pp. 228--229. Theorem 3 gives $f=g+H+S$ with $g$ measurable, and the three parts can be made periodic mod 1. For each $h$ the function $N(h)=\|g(x+h)-g(x)\|_p^p$ is then finite, and it satisfies $N(-h)=N(h)$ and $N(y_1+y_2)\le2^p(N(y_1)+N(y_2))$. As $N$ is measurable, it is below some $K$ on a set of positive measure, and Steinhaus's theorem bounds $N$ near $0$; doubling then bounds $N$ on $[0,1]$. Fubini's theorem gives a point $x$ with $\int_0^1|g(x+y)-g(x)|^p\,dy<\infty$, and periodicity gives $g\in L_p(0,1)$.

Dependencies

Theorem 3 of the paper, and Steinhaus's theorem on difference sets (cited by the paper from its reference [12], p. 145).

Bears on

No Erdős problem page in this corpus; the theorem answers Carroll's question as the paper states it.