Problem 908

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claims/: The 1 claim page of Problem 908, one per claimant's result; the problem's standing derives from them.

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Statement. Let $f:\mathbb{R}\to \mathbb{R}$ be such that $f(x+h)-f(x)$ is measurable for every $h>0$. Is it true that

$$f=g+h+r$$

where $g$ is continuous, $h$ is additive (so $h(x+y)=h(x)+h(y)$), and $r(x+h)-r(x)=0$ for every $h$ and almost all (depending on $h$) $x$?

Statement (corrected). Let $f:\mathbb{R}\to \mathbb{R}$ be such that $f(x+h)-f(x)$ is measurable for every $h>0$. Is it true that

$$f=g+h+r$$

where $g$ is measurable, $h$ is additive (so $h(x+y)=h(x)+h(y)$), and $r(x+h)-r(x)=0$ for every $h$ and almost all (depending on $h$) $x$?

Notes. The change replaces "continuous" by "measurable" as the condition on $g$; nothing else changes. The evidence is the posers' own words. De Bruijn, whom the site credits as co-poser, records the conjecture as Erdős's in [dB51], printed p. 195, "where $g(x)$ is measurable". Erdős's 1982 retrospective [Er82e], Chapter V, §3, printed p. 76, prints "where $g(x)$ is continuous", cites [dB51] and [La80] beside the problem and reports that Laczkovich proved it. Laczkovich proved the measurable form: his Theorem 3 [La80], printed p. 224, proves it, and his introduction, printed p. 217, states Erdős's conjecture with $g$ measurable. The defect is already in [Er82e], and the site's wording follows it. The site's label and the problem's standing judge the corrected Statement. The site also cites [Er81b], Erdős's 'Problems' in The Scottish Book (1981), whose statement of the problem is not recorded here; the correction rests on [dB51], [Er82e] and [La80].

Status. PROVED, the site's label (page last edited 30 December 2025), which describes the corrected Statement: the site's commentary calls the problem a conjecture of de Bruijn and Erdős and credits Laczkovich [La80] with the affirmative answer. Laczkovich's Theorem 3 proves the corrected Statement and is recorded as an accepted full claim.

Source. erdosproblems.com/908, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #908, https://www.erdosproblems.com/908.

References.

• [La80] Laczkovich, M., Functions with measurable differences. Acta Math. Acad. Sci. Hungar. (1980), 217-235. Introduction, printed p. 217; Theorem 3, printed p. 224. Library home: laczkovich_1980_functions_measurable_differences.
• [dB51] de Bruijn, N. G., Functions whose differences belong to a given class. Nieuw Arch. Wiskunde (2) 23 (1951), 194--218. Erdős's conjecture with a measurable summand, printed p. 195. Library home: debruijn_1951_functions_whose_differences_belong_given_class.
• [Er81b] Erdős, P., My Scottish Book 'Problems'. The Scottish Book (1981), 27-35 (page numbers are given for the 2nd edition of The Scottish Book).
• [Er82e] Erdős, P., Some of my favourite problems which recently have been solved. Proceedings of the International Mathematical Conference (Singapore, 1981), North-Holland Math. Stud. 74 (1982), 59--79. Chapter V, §3, printed p. 76. Library home: erdos_1982_my_favourite_problems_which_recently_have.
• [Ke98] Keleti, T., Difference functions of periodic measurable functions. Fund. Math. 157 (1998), 15--32. Theorems 2.9 and 2.13, printed pp. 21--22; Corollary 2.18, printed p. 23. Library home: keleti_1998_difference_functions_periodic_measurable_functions.

Formalization. None recorded.

Current assessment

Laczkovich's Theorem 3 proves the corrected Statement. The theorem is refereed and the site's curator credits it; its proof is not reconstructed in this corpus. The negative-shift identity that carries the positive-shift hypothesis to every shift is elementary and recorded below. Keleti's stronger-hypothesis Theorem 2.13 is recorded below; its general source proofs are uncompiled. No formal verification is recorded.

Progress

Laczkovich's Functions with measurable differences, Acta Math. Acad. Sci. Hungar. 35 (1980), proves the weak difference property for the Lebesgue measurable class $L$ (Theorem 3, printed p.224; introduction p.217). It gives

$$f=g+H+S,$$

with $g$ measurable, $H$ additive, and, for every fixed real $h$, $\Delta_hS=0$ almost everywhere. The exceptional null set may depend on $h$. The hypothesis for negative shifts follows from the positive-shift hypothesis using $\Delta_u f(x)=-\Delta_{-u}f(x+u)$ for $u<0$. This is the corrected Statement. Its proof has not been reconstructed here.

The primary sources differ on the required regularity of $g$. Erdős's Some of my favourite problems which recently have been solved (1982), Chapter V, §3, printed p.76 / physical p.18, states the continuous-$g$ formulation under measurable differences and attributes a proof to Laczkovich. De Bruijn's statement of Erdős's conjecture, Laczkovich's own introduction and Theorem 3 state the measurable-$g$ formulation. The Notes under the corrected Statement record why the continuous summand is a misprint.

Keleti's Difference functions of periodic measurable functions, Fundamenta Mathematicae 157 (1998), gives a separate stronger-hypothesis result. Theorem 2.9 (printed p.21 / physical p.7) says that a measurable function whose every real-shift difference is essentially continuous is itself essentially continuous. Theorem 2.13 (printed p.22 / physical p.8, using the weak-difference definition on p.21) gives the weak difference property for $C^*$, the essentially continuous class. Thus, if every $\Delta_hf$ belongs to $C^*$, there is a decomposition with $g\in C^*$.

Primary sources: Laczkovich 1980, Keleti 1998, and Erdős 1982; the physical and PDF page numbers on this page index the Keleti and Erdős files.

Known Results

The source statements above are recorded at the stated scopes; complete source proofs remain uncompiled. See the source records for Laczkovich's Theorem 3, Keleti's Theorems 2.9 and 2.13 and Corollary 2.18, and de Bruijn's historical formulation.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• debruijn_1951_functions_whose_differences_belong_given_class
• debruijn_1951_functions_whose_differences_belong_given_class / conjecture_p195
• debruijn_1951_functions_whose_differences_belong_given_class / theorem_5_1
• keleti_1998_difference_functions_periodic_measurable_functions
• laczkovich_1980_functions_measurable_differences
• laczkovich_1980_functions_measurable_differences / theorem_3