wrote

In 1884, Scheeffer [4] (pp. 291-293) published a proof of
the following:

Let C be a perfect nowhere dense set of reals and Z be a
countable set of reals. Then for each pair of real numbers
a1 < a2, there exists a real number b such that a1 < b < a2
and the b-translate of C has empty intersection with Z.

For a proof see pp. 52-53 of Young/Young [5] (these page numbers
are for the Chelsea edition). Young/Young's proof actually shows
that the set of real numbers b for which the b-translate of
C contains no points of Z is the complement of a first category
set of real numbers. In particular, there are continuum many such
b's in every open interval.

Boes/Darst/Erdos prove in [2] that given any countable set Z, the
set of b's in (0,1] such that the symmetric Cantor set, formed in
the following manner, is disjoint from Z is the complement of a
first category subset of (0,1]: Remove from [0,1] a segment of
length b/3 so as to leave two intervals of equal length; from
each of these two intervals remove a segment of length b/9 to
leave 4 intervals of equal length; continue in this manner.

In 1959, Bagemihl proved the following stronger version of
Scheeffer's theorem. [See also Morgan [3], pp. 194-195.]

Let F be a first category set of reals and Z be a countable set
of reals. Then the set of real numbers b for which the b-translate
of F contains no points of Z is the complement of a first category
set of real numbers.

[1] Frederick Bagemihl, "A note on Scheeffer's theorem", Michigan
    Math. J. 2 (1954), 149-150.

[2] Duane Boes, Richard Darst, and Paul Erdos, "Fat, Symmetric,
    Irrational Cantor Sets", The American Mathematical Monthly
    88 (1981), 340-341.

[3] John C. Morgan II, POINT SET THEORY, Pure and Applied
    Mathematics 131, Marcel Dekker, 1989. [QA 603 .P67]

[4] L. Scheeffer, "Zur Theorie der stetigen Functionen einer
    reellen Veränderlichen", Acta Math. 5 (1884), 279-296.
    [JFM 16.0340.01 at <http://www.emis.de/MATH/JFM/JFM.html>]

[5] Grace C. Young and William H. Young, THE THEORY OF SETS OF
    POINTS, Cambridge, 1906. [2'nd edition (actually, just a
    revised reprint with an appendix of additional notes) was
    published by Chelsea in 1972, and can be found in most college
    libraries in the QA 248's.]