Is the sum of two countable nowhere dense sets of non-negative real numbers nowhere dense?

[xsfxsf]

(Suggested by a Putnam problem).

A set X of real numbers is nowhere dense if every nonempty open
interval in R contains a nonempty open subinterval disjoint from X.

Let X and Y be countable nowhere dense sets of non-negative real
numbers.

Let X+Y denote {x+y | x \in X and y\in Y}.

Is X+Y necessarily nowhere dense? If so, does this remain true if X
and Y are uncountable?

[G. A. Edgar]

In article <hm13ar$g2i$1...@news.acm.uiuc.edu>, xsfxsf
<henry.te...@gmail.com> wrote:

> (Suggested by a Putnam problem).
>
> A set X of real numbers is nowhere dense if every nonempty open
> interval in R contains a nonempty open subinterval disjoint from X.
>
> Let X and Y be countable nowhere dense sets of non-negative real
> numbers.
>
> Let X+Y denote {x+y | x \in X and y\in Y}.
>
> Is X+Y necessarily nowhere dense?

No. Counterexample suggested by the Cantor set...
Let X = Y = numbers in [0,1) with finite expansions base 3 with only
digits 0 and 2. Then X + Y is dense in [0,2].

> If so, does this remain true if X
> and Y are uncountable?
>

--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/