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Let $D$ be a subset of the decimal digits $ \{ 0, 1, 2, \ldots, 9\}$, with $D \neq \{0\}$ or $\emptyset$. Let $N$ be the set of positive integers whose decimal representations (without leading $0$'s) consist only of digits in $D$. Let $R$ be the set of ratios of numbers in $N$: $R = \{ {x \over y}: x, y \in N\}$. What are the density properties of $R$ in the positive reals?

Clearly, if $D$ contains only one digit, the only accumulation points of $R$ are the powers of $10$ (as well as $0$ and $\infty$, if you consider the compactification of the positive reals), and if $D = \{0,\ldots,9\}$, $R$ is the positive rationals and is dense in the positive reals. Numerical evidence suggests that if $D = \{1,2,3,4\}$, $R$ is dense in the positive reals.

For what sets $D$ is $R$ dense in the positive reals? Generalize for all integral bases $\ge 2$.

Dale
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  • What is your personal work ? – Jean Marie May 08 '19 at 01:47
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    $1/11,10/111,100/1101,1000/11001,\dots$ would seem to converge to $1/11$, in contradiction to your claim about $D={,0,1,}$. – Gerry Myerson May 08 '19 at 02:20
  • And $100/101, 10000/10101, 100000/1010101,\ldots$ converges to $0.99.$ – David K May 08 '19 at 02:30
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    Indeed, the closure of $R$ for ${0,1}$ is uncountable, even if you restrict to denominators that are powers of $10$. – Robert Israel May 08 '19 at 02:30
  • Care to engage with the comments, Dale? – Gerry Myerson May 09 '19 at 03:23
  • Other people have edited my question. One change was to note that $0$ is an accumulation point of any $R$, which I had overlooked because I was only looking for positive accumulation points. Another change was to replace $D={1}$ with $D={0,1}$ on an assertion that the former does not cause $R$ to have accumulation points at the powers of $10$. But unless I am mistaken, an integer of $n+m$ $1$'s divided by an integer of $m$ $1$'s approaches $10^n$ as $m$ goes to infinity. – Dale May 10 '19 at 00:35
  • Maybe you should roll back the offending edit, Dale. – Gerry Myerson May 10 '19 at 04:39
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    I've updated the question to correct the bit about when $R$ has accumulation points at the powers of 10. I've also expanded the description a bit to clarify attention toward the positive reals and that $0$ and $\infty$ are elements of the compatification thereof. – Dale May 18 '19 at 12:38

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