Generalized Cantor Expansions, Joseph Galante
Referee Report
This paper contains some interesting mathematics, and is certainly at the appropriate
level for the Rose-Hulman Undergraduate Mathematics Journal. However, before
publication, several issues must be addressed by the author. Some of the issues pertain to
improving formatting. However there are some errors in reasoning caused by poor
language and insufficient explanations.
The organization would be greatly helped by the use of section numbers and, more
importantly, theorem numbers, lemma numbers, and equation numbers for the important
equations. Phrases such as “the theorem” or “the identity” should be replaced by
“Theorem 1” or “Identity (1.1)”, for example.
The remaining comments are by page.
Page 1. Provide a reference for the proof of the uniqueness of the Cantor expansion of a
natural number.
Page 2. S is not just a set, it’s an ordered set, or sequence. This should be noted. Both
p(n) and x_n refer to the same thing. Is this redundant? In the base case of the induction
proof, the proof begins with what we want to show and concludes with what we know.
This is backwards logic. The phrase “Assume that this works for …” is an example of
where an equation number would be useful for cross-referencing.
Page 3. In the middle of the page appears the word “lets” which should be a contraction
or “let us.” The proof of Case 1 requires much more explanation. For starters, it is
unclear what “this number” refers to in the second paragraph. In the equation beginning
with a_i*P(i), an explanation is necessary. It seems that this is where the hypothesis of
the case is being used, but how it is being used needs to be explained. It seems unclear
why this works for any such i. Below that, it is unclear which terms are being denoted by
the ellipsis.
Page 4. The paragraph beginning with “Note that …” is strangely phrased.
Page 5. Be consistent with where the primes are placed. The phrase “linearly
independent of each other” seems inappropriate. Is this true linear independence? This
does not explain what you’re concluding. In particular, a_i-a’_i could possibly be
negative. There is a big logical jump here that must be explained. In one of the
examples, a 137 should be changed to a 138.
Page 6. In the first full paragraph, the phrase “… base used in the example…” would be
helped by using numbering. Or say the “above example”. The statement that there are
uncountably many GCEs is vague. Does that mean of one fixed number? “Expansion”
usually refers to something with respect to a particular expression. I believe that you
mean that there are uncountable many choices for S. Your argument only justifies that
the cardinality of the set of all possible S’s is greater than or equal to the cardinality of
the reals. There is more work to prove them equal. In addition, if you desire to denote
the cardinality of the reals, you should use c and not aleph_1.
Page 7. Carat symbols should be avoided for exponents. Because S is an ordered set,
writing S={1} u {primes} is inappropriate without mentioning that the primes are taken
in order.
Page 8. “Primorials” seems like a term that deserves a reference. The base case of the
proof by induction starts with what we want to show and concludes what we already
know. This should be reversed.
Page 9. In the fourth line of the initial list of equalities, involving both division and
multiplication, parentheses should be used to clear up the ordering. This proof of the
inductive step also begins with what is trying to be shown and going backwards. The
theorem at the bottom of the page states that P is convergent. Since you are avoiding
typical sequence language, you should remark that convergence is as n tends to infinity.
Page 10. The proof of the theorem begins with “By the lemma” but what you are quoting
is not given as a lemma. State it as a lemma and use a number. The very first line of
equations shows an increasing sequence of positive numbers bounded above. This is
sufficient to conclude convergence. The remaining argument seems unnecessary. The
theorem stated at the bottom of the page is not well-stated. Perhaps you mean “with
respect to a given S.”
Page 11. At the bottom of page 12 is a lemma about subdividing [0,1) using terminating
FGCEs. This seems to be the idea that justifies the first equation at the top of this page.
It is unclear why the density of Q in R is relevant. In addition, the first equation depends
upon an initial choice of n which does not appear. More instruction should be given in
how the c_i’s are chosen. The statement “We want this true…” does not mean that it is.
That still needs to be justified. In the epsilon-N statement, a 1 should be replaced with an
x. I believe employing the Squeeze theorem would simplify this proof significantly.
In the explanation of the lack of uniqueness of decimal expansions, it is said that “This
can be true for any real number.” Is it? The answer is known. What does it mean for
two things to differ by an arbitrarily small amount? Doesn’t that mean that they’re
equal? The FGCE’s are infinite series that converge to the same value even if the
coefficients differ. I don’t really understand what the point of the theorem is.
Page 12. Proper FGCE is defined using language that implies that it always exists, given
a particular x. That is not true. The lemma at the bottom of the page is imprecisely
phrased. You seem to want all FGCEs that contain only terms up to and including P(n).
This should be incorporated into the statement of the theorem to tighten it up.
Page 13. In the first big paragraph, the phrase “By a previous theorem…” really needs
theorem numbering for cross-referencing. The quantity l (ell) has two different meanings
in the proof. This proof would be much better organized if we first choose n, then
choose m between 0 and P(n)-1, and then show that there is an appropriate FGCE equal
to m/P(n). The proof, as is, is flawed.
In the second proof, the phrase “By a previous lemma…” requires numbering. In
addition, the lemma quoted depends upon choosing i first. This should be pointed out.
Page 14. When you conclude that b divides x_1…x_n, the explanation is invalid. Two
fractions have been shown to be equal, but that does not imply that one denominator
divides the other. Perhaps a and b should be relatively prime. Even then, a better
explanation of this crucial point would be illuminating. In the sentence beginning with
“For example consider…” the existence of i depends upon an initial choice of b. This
should be rephrased or omitted. In the paragraph preceding the example at the bottom of
the page, it is stated that S is uncountably infinite. This is false, S is a sequence, which is
countable. I believe the desired point is that there are uncountable many choices for S.
Page 15. The quote of Knuth should be cited.