On a theorem of van der Corput

Approximate homomorphisms

“…Also J. P. R. Christensen (1972) proved (8.3) for the case in which A is a Christensen measurable subset of an Abelian Polish group, where A is not a Christensen zero set. These remarks lead to the following results (see K. Baron (1985), K. Baron and P. Volkmann (1988) For the case where E is a real linear topologial Baire space, the same statement holds if we replace the word "Lebesgue" by "Baire". In the case where E is a separable real F-space the statement is true if we replace the word "Lebesgue" by "Christensen".…”

“…In the second corollary the hypotheses are that f: E~R satisfies (8.1), that there is a set A c E and an e > 0 such that the set A -A has a nonempty interior and f(A) ~ ( -1/4 + e, 1/4 --e) + Z, with the same conclusion. This second corollary is of use in connection with measurability requirements on f, as indicated in Remark 1 of K. Baron and P. Volkmann (1988). A theorem of Steinhaus (1920, Theorem VIII) states that any set A ~ R of positive Lebesgue measure satisfies the condition…”

“…Where Baron and Volkmann consider functions measurable along rays, Baker deals with characters which are continuous along rays. Theorems of Baron (1985), Baron and Volkmann (1988) involving measurability hypotheses are obtained by using a known theorem AEQ. MATH.…”

On approximately additive functions

“…Our results correspond to the problem of Hyers-Ulam stability for the Cauchy equation (see e.g. [6,17,18,22] and [27,Chapter VI]) and to the subjects considered, e.g., in [2,4,5,8,19,28], where functions satisfying (1), with C = {0}, have been investigated (actually we generalize several results from those papers).…”

On functions having Cauchy differences in some prescribed sets

“…This theorem was extended in [3] for real functionals. Example 2 shows that the above quoted theorem of van der Corput cannot be extended to functions taking values in ~2 (where 7/would be replaced by a discrete subgroup of ~2).…”