On Ditkin sets
In the study of spectral synthesis S-sets and C-sets (see Rudin ; Reiter  uses the terminology Wiener sets and Wiener-Ditkin sets respectively) have been discussed extensively. A new concept of Ditkin sets was introduced and studied by Stegeman in  so that, in Reiter's terminology, Wiener-Ditkin sets are precisely sets which are both Wiener sets and Ditkin sets. The importance of such sets in spectral synthesis and their connection to the C-set-S-set problem (see Rudin ) are mentioned there. In this paper we study local properties, unions and intersections of Ditkin sets. (Warning: Usually in the literature "Ditkin set" means "C-set", but we follow the terminology of Stegeman.) Our results include: (i) if each point of a closed set E has a closed relative Ditkin neighbourhood, then E is a Ditkin set; (ii) any closed countable union of Ditkin sets is a Ditkin set; (iii) if $E_1 ∩ E_2$ is a Ditkin set, then $E_1 ∩ E_2$ is a Ditkin set if and only if $E_1$ and $E_2$ are Ditkin sets; and (iv) if $E_1, E_2$ are Ditkin sets with disjoint boundaries then $E_1 ∩ E_2$ is a Ditkin set.
-  T. K. Muraleedharan and K. Parthasarathy, On unions and intersections of sets of synthesis, Proc. Amer. Math. Soc., to appear.
-  H. Reiter, Classical Harmonic Analysis and Locally Compact Groups, Oxford University Press, Oxford, 1968.
-  W. Rudin, Fourier Analysis on Groups, Interscience, New York, 1962.
-  J. D. Stegeman, Some problems on spectral synthesis, in: Proc. Harmonic Analysis (Iraklion, 1978), Lecture Notes in Math. 781, Springer, Berlin, 1980, 194-203.