Don Reble <djr@nk.ca> wrote:

See OEIS A110081.

That lists "Integers n such that the digit set D = (0, 1, -n) used in base 3 expansions of the form Sum_{ -N < j < infty} d_j 3^{-j}, all d_j in D, can represent all real numbers." It begins 1, 7, 25, 31... But that doesn't agree with my results. I concur on 1 and 7. I didn't test anything as high as 25. But please name a real number which can't be represented by {0, 1, -4}. Or by {0, 1, -10}. I can't find one. I've confirmed that all integers up to at least 300 can be represented with a 7-digit ternary number using those digits. (Larger integers presumably simply require more digits.) (Real numbers can of course be represented by moving the radix point to the left. Move it N places to the left and you can approximate any real number to a precision of 3^-N.) But neither have I proven there isn't one. Without such a proof, I'm reluctant to update OEIS. I'll look for a proof based on finding patterns in the digits. "Lucas, Stephen K - lucassk" <lucassk@jmu.edu> wrote:

Considering the fact that Matula?s paper came out in 1982, but was based upon a technical report dating to 1978, I wonder who should get priority for discovering digit sets that are not contiguous and still can be used to represent integers and reals. Odlyzko (first page is available) suggests a question by Knuth.

I'm surprised it's so recent. It's the sort of thing that Fibonacci could have done.

And another fun result: most papers I?ve read on the topic include 0 as a digit. It turns out not to be necessary. In https://www.math.tugraz.at/fosp/pdfs/tugraz_0024.pdf they prove that in base 2 you can use the digit set {1,4} except for the leading digit, which may be {1,2,4}

Interesting. I've been looking only at sets of N digits for base N. Even so, I was mistaken when I said ternary had to have 0. I tried all seven-digit ternary numbers using each possible sets of three digits from -9 through 9. I now realize I was implicitly thinking that seven-digit numbers included all shorter numbers too, since the leading digits can be zeros. Well, no, the leading digits *can't* be zeros if there's no zero in the set! I've corrected that, and discovered 32 sets of three digits not containing 0 to go with the 9 sets I had previously discovered that did contain 0. For instance {-3, 1, 2}. Here are zero through ten in that form of ternary, with 3 meaning -3: 13, 1, 2, 23, 11, 12, 233, 21, 22, 113, 231. Also, any number can be prefaced with 13 without changing its value. Hence 1 = 131 = 13131 = 1313131, etc.