Remarks on two problems by M. Laczkovich on functions with Borel measurable differences (Q955175)

The author considers two problems asked by \textit{M.~Laczkovich} [Acta Math. Acad. Sci. Hungar. 35, 217--235 (1980; Zbl 0468.28006)] concerning the difference property of Borel measurable functions: Problem~2: For \(f: \mathbb R\to \mathbb R\) and \(h\in \mathbb R\) set \(\Delta_h f=f(x+h)-f(x)\). Suppose that \(\Delta_h f\) is Borel measurable for each \(h\). Is it true that all \(\Delta_h f\) belong to the Baire class \(\alpha\) for some \(\alpha<\omega_1\)? Problem~3: Let \(f\) be Borel measurable such that all \(\Delta_h f\) are of Baire class \(\alpha\). Does it follow that \(f\) is of class \(\alpha\), too? The author proves the following theorems: (1) If \(\alpha>0\) is a countable ordinal and \(f:\mathbb R\to \mathbb R\) is a Borel function such that \(\Delta_h f\) is \(\Sigma_{\alpha}^0\) measurable for every \(h\in \mathbb R\), then \(f\) is \(\Sigma^0_{1+\alpha}\) measurable. This affirmatively answers Problem~3 for all infinite \(\alpha\). The problem for finite \(\alpha>1\) remains open. (Recall that for all bounded Borel functions this problem has been solved affirmatively by \textit{M.~Laczkovich} [in: G. Halász et al. (eds.), Paul Erdős and his mathematics I. Based on the conference, Budapest, Hungary, July 4--11, 1999. Berlin: Springer. Bolyai Soc. Math. Stud. 11, 363--410 (2002; Zbl 1035.39016)].) (2) Assume the measure uniformization principle. Let \(f:\mathbb R\to \mathbb R\). If \(\Delta_h f\) is Borel for almost every \(h\), then \(f\) is Borel. Moreover, if the Lebesgue measure is \(\omega_2\)-additive, every projective set is measurable, and \(f:\mathbb R\to \mathbb R\) is a projective function with all \(\Delta_h f\) Borel, then there is a countable ordinal \(\alpha\) such that \(\Delta_h f\) is \(\Sigma^0_{\alpha}\) measurable for all \(h\). In particular, Problem~3 has an affirmative answer under the axiom of determinacy or in the Solovay model. Recall that \textit{R.~Filipów} and \textit{I.~Recław} [Acta Math. Hung. 96, 21--25 (2002; Zbl 1006.28006)] showed that CH implies the negative answer to Problem~3.