Conjecture 1 (Collatz conjecture) For any given natural number , the orbit passes through (i.e. for some ).

Open questions with this level of notoriety can lead to what Richard Lipton calls “mathematical diseases” (and what I termed an unhealthy amount of obsession on a single famous problem). (See also this xkcd comic regarding the Collatz conjecture.) As such, most practicing mathematicians tend to spend the majority of their time on more productive research areas that are only just beyond the range of current techniques. Nevertheless, it can still be diverting to spend a day or two each year on these sorts of questions, before returning to other matters; so I recently had a go at the problem. Needless to say, I didn’t solve the problem, but I have a better appreciation of why the conjecture is (a) plausible, and (b) unlikely be proven by current technology, and I thought I would share what I had found out here on this blog.

Let me begin with some very well known facts. If is odd, then is even, and so . Because of this, one could replace by the function , defined by when is odd, and when is even, and obtain an equivalent conjecture. Now we see that if one chooses “at random”, in the sense that it is odd with probability and even with probability , then increases by a factor of roughly half the time, and decreases it by a factor of half the time. Furthermore, if is uniformly distributed modulo , one easily verifies that is uniformly distributed modulo , and so should be roughly times as large as half the time, and roughly times as large as the other half of the time. Continuing this at a heuristic level, we expect generically that half the time, and the other half of the time. The logarithm of this orbit can then be modeled heuristically by a random walk with steps and occuring with equal probability. The expectation

is negative, and so (by the classic gambler’s ruin) we expect the orbit to decrease over the long term. This can be viewed as heuristic justification of the Collatz conjecture, at least in the “average case” scenario in which is chosen uniform at random (e.g. in some large interval ). (It also suggests that if one modifies the problem, e.g. by replacing to , then one can obtain orbits that tend to increase over time, and indeed numerically for this variant one sees orbits that appear to escape to infinity.) Unfortunately, one can only rigorously keep the orbit uniformly distributed modulo for time about or so; after that, the system is too complicated for naive methods to control at anything other than a heuristic level.

Remark 1 One can obtain a rigorous analogue of the above arguments by extending from the integers to the -adics . This compact abelian group comes with a Haar probability measure, and one can verify that this measure is invariant with respect to ; with a bit more effort one can verify that it is ergodic. This suggests the introduction of ergodic theory methods. For instance, using the pointwise ergodic theorem, we see that if is a random -adic integer, then almost surely the orbit will be even half the time and odd half the time asymptotically, thus supporting the above heuristics. Unfortunately, this does not directly tell us much about the dynamics on , as this is a measure zero subset of . More generally, unless a dynamical system is somehow “polynomial”, “nilpotent”, or “unipotent” in nature, the current state of ergodic theory is usually only able to say something meaningful about generic orbits, but not about all orbits. For instance, the very simple system on the unit circle is well understood from ergodic theory (in particular, almost all orbits will be uniformly distributed), but the orbit of a specific point, e.g. , is still nearly impossible to understand (this particular problem being equivalent to the notorious unsolved question of whether the digits of are uniformly distributed).

The above heuristic argument only suggests decreasing orbits for almost all (though even this remains unproven, the state of the art is that the number of in that eventually go to is , a result of Krasikov and Lagarias). It leaves open the possibility of some very rare exceptional for which the orbit goes to infinity, or gets trapped in a periodic loop. Since the only loop that lies in is (for ) or (for ), we thus may isolate a weaker consequence of the Collatz conjecture:

Conjecture 2 (Weak Collatz conjecture) Suppose that is a natural number such that for some . Then is equal to , , or .

Of course, we may replace with (and delete ““) and obtain an equivalent conjecture.

This weaker version of the Collatz conjecture is also unproven. However, it was observed by Bohm and Sontacchi that this weak conjecture is equivalent to a divisibility problem involving powers of and :

Conjecture 3 (Reformulated weak Collatz conjecture) There does not exist and integers

such that is a positive integer that is a proper divisor of

i.e.

for some natural number .

Proposition 4 Conjecture 2 and Conjecture 3 are equivalent.

Proof: To see this, it is convenient to reformulate Conjecture 2 slightly. Define an equivalence relation on by declaring if for some integer , thus giving rise to the quotient space of equivalence classes (which can be placed, if one wishes, in one-to-one correspondence with the odd natural numbers). We can then define a function by declaring

for any , where is the largest power of that divides . It is easy to see that is well-defined (it is essentially the Syracuse function, after identifying with the odd natural numbers), and that periodic orbits of correspond to periodic orbits of or . Thus, Conjecture 2 is equivalent to the conjecture that is the only periodic orbit of .

Now suppose that Conjecture 2 failed, thus there exists such that for some . Without loss of generality we may take to be odd, then . It is easy to see that is the only fixed point of , and so . An easy induction using (2) shows that

where, for each , is the largest power of that divides

In particular, as is odd, . Using the recursion

we see from induction that divides , and thus :

Since , we have

for some integer . Since is divisible by , and is odd, we conclude ; if we rearrange the above equation as (1), then we obtain a counterexample to Conjecture 3.

Conversely, suppose that Conjecture 3 failed. Then we have , integers

and a natural number such that (1) holds. As , we see that the right-hand side of (1) is odd, so is odd also. If we then introduce the natural numbers by the formula (3), then an easy induction using (4) shows that

with the periodic convention for . As the are increasing in (even for ), we see that is the largest power of that divides the right-hand side of (5); as is odd, we conclude that is also the largest power of that divides . We conclude that

and thus is a periodic orbit of . Since is an odd number larger than , this contradicts Conjecture 3.

Call a counterexample a tuple that contradicts Conjecture 3, i.e. an integer and an increasing set of integers

such that (1) holds for some . We record a simple bound on such counterexamples, due to Terras and to Garner :

Lemma 5 (Exponent bounds) Let , and suppose that the Collatz conjecture is true for all . Let be a counterexample. Then

Proof: The first bound is immediate from the positivity of . To prove the second bound, observe from the proof of Proposition 4 that the counterexample will generate a counterexample to Conjecture 2, i.e. a non-trivial periodic orbit . As the conjecture is true for all , all terms in this orbit must be at least . An inspection of the proof of Proposition 4 reveals that this orbit consists of steps of the form , and steps of the form . As all terms are at least , the former steps can increase magnitude by a multiplicative factor of at most . As the orbit returns to where it started, we conclude that

whence the claim.

The Collatz conjecture has already been verified for many values of (up to at least , according to this web site). Inserting this into the above lemma, one can get lower bounds on . For instance, by methods such as this, it is known that any non-trivial periodic orbit has length at least , as shown in Garner’s paper (and this bound, which uses the much smaller value that was available in 1981, can surely be improved using the most recent computational bounds).

Now we can perform a heuristic count on the number of counterexamples. If we fix and , then , and from basic combinatorics we see that there are different ways to choose the remaining integers

to form a potential counterexample . As a crude heuristic, one expects that for a “random” such choice of integers, the expression (1) has a probability of holding for some integer . (Note that is not divisible by or , and so one does not expect the special structure of the right-hand side of (1) with respect to those moduli to be relevant. There will be some choices of where the right-hand side in (1) is too small to be divisible by , but using the estimates in Lemma 5, one expects this to occur very infrequently.) Thus, the total expected number of solutions for this choice of is

The heuristic number of solutions overall is then expected to be

where, in view of Lemma 5, one should restrict the double summation to the heuristic regime , with the approximation here accurate to many decimal places.

We need a lower bound on . Here, we will use Baker’s theorem (as discussed in this previous post), which among other things gives the lower bound

for some absolute constant . Meanwhile, Stirling’s formula (as discussed in this previous post) combined with the approximation gives

where is the entropy function

A brief computation shows that

and so (ignoring all subexponential terms)

which makes the series (6) convergent. (Actually, one does not need the full strength of Lemma 5 here; anything that kept well away from would suffice. In particular, one does not need an enormous value of ; even (say) would be more than sufficient to obtain the heuristic that there are finitely many counterexamples.) Heuristically applying the Borel-Cantelli lemma, we thus expect that there are only a finite number of counterexamples to the weak Collatz conjecture (and inserting a bound such as , one in fact expects it to be extremely likely that there are no counterexamples at all).

This, of course, is far short of any rigorous proof of Conjecture 2. In order to make rigorous progress on this conjecture, it seems that one would need to somehow exploit the structural properties of numbers of the form

In some very special cases, this can be done. For instance, suppose that one had with at most one exception (this is essentially what is called a -cycle by Steiner). Then (8) simplifies via the geometric series formula to a combination of just a bounded number of powers of and , rather than an unbounded number. In that case, one can start using tools from transcendence theory such as Baker’s theorem to obtain good results; for instance, in the above-referenced paper of Steiner, it was shown that -cycles cannot actually occur, and similar methods have been used to show that -cycles (in which there are at most exceptions to ) do not occur for any , as was shown by Simons and de Weger. However, for general increasing tuples of integers , there is no such representation by bounded numbers of powers, and it does not seem that methods from transcendence theory will be sufficient to control the expressions (8) to the extent that one can understand their divisibility properties by quantities such as .

Amusingly, there is a slight connection to Littlewood-Offord theory in additive combinatorics – the study of the random sums

generated by some elements of an additive group , or equivalently, the vertices of an -dimensional parallelepiped inside . Here, the relevant group is . The point is that if one fixes and (and hence ), and lets vary inside the simplex

then the set of all sums of the form (8) (viewed as an element of ) contains many large parallelepipeds. (Note, incidentally, that once one fixes , all the sums of the form (8) are distinct; because given (8) and , one can read off as the largest power of that divides (8), and then subtracting off one can then read off , and so forth.) This is because the simplex contains many large cubes. Indeed, if one picks a typical element of , then one expects (thanks to Lemma 5) that there there will be indices such that for , which allows one to adjust each of the independently by if desired and still remain inside . This gives a cube in of dimension , which then induces a parallelepiped of the same dimension in . A short computation shows that the generators of this parallelepiped consist of products of a power of and a power of , and in particular will be coprime to .

If the weak Collatz conjecture is true, then the set must avoid the residue class in . Let us suppose temporarily that we did not know about Baker’s theorem (and the associated bound (7)), so that could potentially be quite small. Then we would have a large parallelepiped inside a small cyclic group that did not cover all of , which would not be possible for small enough. Indeed, an easy induction shows that a -dimensional parallelepiped in , with all generators coprime to , has cardinality at least . This argument already shows the lower bound . In other words, we have

Proposition 6 Suppose the weak Collatz conjecture is true. Then for any natural numbers with , one has .

This bound is very weak when compared against the unconditional bound (7). However, I know of no way to get a nontrivial separation property between powers of and powers of other than via transcendence theory methods. Thus, this result strongly suggests that any proof of the Collatz conjecture must either use existing results in transcendence theory, or else must contribute a new method to give non-trivial results in transcendence theory. (This already rules out a lot of possible approaches to solve the Collatz conjecture.)

By using more sophisticated tools in additive combinatorics, one can improve the above proposition (though it is still well short of the transcendence theory bound (7)):

Proposition 7 Suppose the weak Collatz conjecture is true. Then for any natural numbers with , one has for some absolute constant .

Proof: (Informal sketch only) Suppose not, then we can find with of size . We form the set as before, which contains parallelepipeds in of large dimension that avoid . We can count the number of times occurs in one of these parallelepipeds by a standard Fourier-analytic computation involving Riesz products (see Chapter 7 of my book with Van Vu, or this recent preprint of Maples). Using this Fourier representation, the fact that this parallelepiped avoids (and the fact that ) forces the generators to be concentrated in a Bohr set, in that one can find a non-zero frequency such that of the generators lie in the set . However, one can choose the generators to essentially have the structure of a (generalised) geometric progression (up to scaling, it resembles something like for ranging over a generalised arithmetic progression, and a fixed irrational), and one can show that such progressions cannot be concentrated in Bohr sets (this is similar in spirit to the exponential sum estimates of Bourgain on approximate multiplicative subgroups of , though one can use more elementary methods here due to the very strong nature of the Bohr set concentration (being of the “ concentration” variety rather than the “ concentration”).). This furnishes the required contradiction.

Thus we see that any proposed proof of the Collatz conjecture must either use transcendence theory, or introduce new techniques that are powerful enough to create exponential separation between powers of and powers of .

Unfortunately, once one uses the transcendence theory bound (7), the size of the cyclic group becomes larger than the volume of any cube in , and Littlewood-Offord techniques are no longer of much use (they can be used to show that is highly equidistributed in , but this does not directly give any way to prevent from containing ).

One possible toy model problem for the (weak) Collatz conjecture is a conjecture of Erdos asserting that for , the base representation of contains at least one . (See this paper of Lagarias for some work on this conjecture and on related problems.) To put it another way, the conjecture asserts that there are no integer solutions to

with and . (When , of course, one has .) In this form we see a resemblance to Conjecture 3, but it looks like a simpler problem to attack (though one which is still a fair distance beyond what one can do with current technology). Note that one has a similar heuristic support for this conjecture as one does for Proposition 3; a number of magnitude has about base digits, so the heuristic probability that none of these digits are equal to is , which is absolutely summable.

Theorem 1 (Gelfond-Schneider theorem) Let be algebraic numbers, with and irrational. Then (any of the values of the possibly multi-valued expression) is transcendental.

For sake of simplifying the discussion, let us focus on just one specific consequence of this theorem:

Corollary 2 is transcendental.

Proof: If not, one could obtain a contradiction to the Gelfond-Schneider theorem by setting and . (Note that is clearly irrational, since for any integers with positive.)

In the 1960s, Alan Baker established a major generalisation of the Gelfond-Schneider theorem known as Baker’s theorem, as part of his work in transcendence theory that later earned him a Fields Medal. Among other things, this theorem provided explicit quantitative bounds on exactly how transcendental quantities such as were. In particular, it gave a strong bound on how irrational such quantities were (i.e. how easily they were approximable by rationals). Here, in particular, is one special case of Baker’s theorem:

Proposition 3 (Special case of Baker’s theorem) For any integers with positive, one has

for some absolute (and effectively computable) constants .

This theorem may be compared with (the easily proved) Liouville’s theorem on diophantine approximation, which asserts that if is an irrational algebraic number of degree , then

for all with positive, and some effectively computable , and (the more significantly difficult) Thue-Siegel-Roth theorem, which under the same hypotheses gives the bound

for all , all with positive and an ineffective constant . Finally, one should compare these results against Dirichlet’s theorem on Diophantine approximation, which asserts that for any real number one has

for infinitely many with positive. (The reason the Thue-Siegel-Roth theorem is ineffective is because it relies heavily on the dueling conspiracies argument, i.e. playing off multiple “conspiracies” against each other; the other results however only focus on one approximation at a time and thus avoid ineffectivity.)

Proposition 3 easily implies the following separation property between powers of and powers of :

Corollary 4 (Separation between powers of and powers of ) For any positive integers one has

for some effectively computable constants (which may be slightly different from those in Proposition 3).

Indeed, this follows quickly from Proposition 3, the identity

and some elementary estimates.

In particular, the gap between powers of three and powers of two grows exponentially in the exponents . I do not know of any other way to establish this fact other than essentially going through some version of Baker’s argument (which will be given below).

For comparison, by exploiting the trivial (yet fundamental) integrality gap – the obvious fact that if an integer is non-zero, then its magnitude is at least – we have the trivial bound

for all positive integers (since, from the fundamental theorem of arithmetic, cannot vanish). Putting this into (1) we obtain a very weak version of Proposition 3, that only gives exponential bounds instead of polynomial ones:

Proposition 5 (Trivial bound) For any integers with positive, one has

for some absolute (and effectively computable) constant .

The proof of Baker’s theorem (or even of the simpler special case in Proposition 3) is largely elementary (except for some very basic complex analysis), but is quite intricate and lengthy, as a lot of careful book-keeping is necessary in order to get a bound as strong as that in Proposition 3. To illustrate the main ideas, I will prove a bound that is weaker than Proposition 3, but still significantly stronger than Proposition 5, and whose proof already captures many of the key ideas of Baker:

Proposition 6 (Weak special case of Baker’s theorem) For any integers with , one has

for some absolute constants .

Note that Proposition 3 is equivalent to the assertion that one can take (and effective) in the above proposition.

The proof of Proposition 6 can be made effective (for instance, it is not too difficult to make the close to ); however, in order to simplify the exposition (and in particular, to use some nonstandard analysis terminology to reduce the epsilon management), I will establish Proposition 6 with ineffective constants .

Like many other results in transcendence theory, the proof of Baker’s theorem (and Proposition 6) rely on what we would nowadays call the polynomial method – to play off upper and lower bounds on the complexity of polynomials that vanish (or nearly vanish) to high order on a specified set of points. (I have discussed the polynomial method in relation to problems in incidence geometry in several previous blog posts.) In the specific case of Proposition 6, the points in question are of the form

for some large integer . On the one hand, the irrationality of ensures that the curve

is not algebraic, and so it is difficult for a polynomial of controlled complexity to vanish (or nearly vanish) to high order at all the points of ; the trivial bound in Proposition 5 allows one to make this statement more precise. (Here, “complexity” of a polynomial is an informal term referring both to the degree of the polynomial, and the height of the coefficients, which in our application will essentially be integers up to some normalisation factors.) On the other hand, if Proposition 6 failed, then is close to a rational, which by Taylor expansion makes close to an algebraic curve over the rationals (up to some rescaling by factors such as and ) at each point of . This, together with a pigeonholing argument, allows one to find a polynomial of reasonably controlled complexity to (nearly) vanish to high order at every point of .

These observations, by themselves, are not sufficient to get beyond the trivial bound in Proposition 5. However, Baker’s key insight was to exploit the integrality gap to bootstrap the (near) vanishing of on a set to imply near-vanishing of on a larger set with . The point is that if a polynomial of controlled degree and size (nearly) vanishes to higher order on a lot of points on an analytic curve such as , then it will also be fairly small on many other points in as well. (To quantify this statement efficiently, it is convenient to use the tools of complex analysis, which are particularly well suited to understand zeroes (or small values) of polynomials.) But then, thanks to the integrality gap (and the controlled complexity of ), we can amplify “fairly small” to “very small”.

Using this observation and an iteration argument, Baker was able to take a polynomial of controlled complexity that nearly vanished to high order on a relatively small set , and bootstrap that to show near-vanishing on a much larger set . This bootstrap allows one to dramatically bridge the gap between the upper and lower bounds on the complexity of polynomials that nearly vanish to a specified order on a given , and eventually leads to Proposition 6 (and, with much more care and effort, to Proposition 3).

Below the fold, I give the details of this argument. My treatment here is inspired by this expose of Serre, and these lecture notes of Soundararajan (as transcribed by Ian Petrow).

— 1. Nonstandard formulation —

The proof of Baker’s theorem requires a lot of “epsilon management” in that one has to carefully choose a lot of parameters such as and in order to make the argument work properly. This is particularly the case if one wants a good value of exponents in the final result, such as the quantity in Proposition 6. To simplify matters, we will abandon all attempts to get good values of constants anywhere, which allows one to retreat to the nonstandard analysis setting where the notation is much cleaner, and much (though not all) of the epsilon management is eliminated. This is a relatively mild use of nonstandard analysis, though, and it is not difficult to turn all the arguments below into standard effective arguments (but at the cost of explicitly tracking all the constants ). See for instance the notes of Soundararajan for such an effective treatment.

We turn to the details. We will assume some basic familiarity with nonstandard analysis, as covered for instance in this previous blog post (but one should be able to follow this argument using only non-rigorous intuition of what terms such as “unbounded” or “infinitesimal” mean).

Let be an unbounded (nonstandard) positive real number. Relative to this , we can define various notions of size:

• A nonstandard number is said to be of polynomial size if one has for some standard .
• A nonstandard number is said to be of polylogarithmic size if one has for some standard .
• A nonstandard number is said to be of quasipolynomial size if one has for some standard .
• A nonstandard number is said to be quasiexponentially small if one has for every standard .
• Given two nonstandard numbers with non-negative, we write or if for some standard . We write or if we have for all standard .

As a general rule of thumb, in our analysis all exponents will be of polylogarithmic size, all coefficients will be of quasipolynomial size, and all error terms will be quasiexponentially small.

In this nonstandard analysis setting, there is a clean calculus (analogous to the calculus of the asymptotic notations and ) to manipulate these sorts of quantities without having to explicitly track the constants . For instance:

• The sum, product, or difference of two quantities of a given size (polynomial, polylogarithmic, quasipolynomial, or quasiexponentially small) remains of that given size (i.e. each size range forms a ring).
• If , and is of a given size, then is also of that size.
• If is of quasipolynomial size and is of polylogarithmic size, then is of quasipolynomial size, and (if is a natural number) is also of quasipolynomial size.
• If is quasiexponentially small, and is of quasipolynomial size, then is also quasiexponentially small. (Thus, the quasiexponentially small numbers form an ideal in the ring of quasipolynomial numbers.)
• Any quantity of polylogarithmic size, is of polynomial size; and any quantity of polynomial size, is of quasipolynomial size.

We will refer to these sorts of facts as asymptotic calculus, and rely upon them heavily to simplify a lot of computations (particularly regarding error terms).

Proposition 6 is then equivalent to the following assertion:

Proposition 7 (Nonstandard weak special case of Baker) Let be an unbounded nonstandard natural number, and let be a rational of height at most (i.e. ). Then is not quasiexponentially small (relative to , of course).

Let us quickly see why Proposition 7 implies Proposition 6 (the converse is easy and is left to the reader). This is the usual “compactness and contradiction” argument. Suppose for contradiction that Proposition 6 failed. Carefully negating the quantifiers, we may then find a sequence of (standard) rationals with , such that

for all natural numbers . As is irrational, must go to infinity. Taking the ultralimit of the , and setting to be (say) , we contradict Proposition 7.

It remains to prove Proposition 7. We fix the unbounded nonstandard natural number , and assume for contradiction that is quasiexponentially close to a nonstandard rational of height at most . We will write for the assertion that is quasiexponentially small, thus

The objective is to show that (2) leads to a contradiction.

— 2. The polynomial method —

Now it is time to introduce the polynomial method. We will be working with the following class of polynomials:

Definition 8 A good polynomial is a nonstandard polynomial of the form

of two nonstandard variables of some (nonstandard) degree at most (in each variable), where is a nonstandard natural number of polylogarithmic size, and whose coefficients are (nonstandard) integers of quasipolynomial size. (A technical point: we require the to depend in an internal fashion on the indices , in order for the nonstandard summation here to be well-defined.) Define the height of the polynomial to be the maximum magnitude of the coefficients in ; thus, by hypothesis, is of quasipolynomial size.

We have a key definition:

Definition 9 Let be two (nonstandard) positive numbers of polylogarithmic size. A good polynomial is said to nearly vanish to order on if one has

for all nonstandard natural numbers and .

The derivatives in (4) can be easily computed. Indeed, if we expand out the good polynomial out as (3), then the left-hand side of (4) is

Now, from (2) we have

Using the asymptotic calculus (and the hypotheses that are of polylogarithmic size, and the are of quasipolynomial size) we conclude that the left-hand side of (4) is

The quantity (and its reciprocal) is of quasipolynomial size. Thus, the condition (4) is equivalent to the assertion that

for all and ; as the left-hand side is a nonstandard integer, we see from the integrality gap that the condition is in fact equivalent to the exact constraint

for all and .

Using this reformulation of (4), we can now give some upper and lower bounds on the complexity of good polynomials that nearly vanish to a high order on a set . We first give an lower bound, that prevents the degree from being smaller than :

Proposition 10 (Lower bound) Let be a non-trivial good polynomial of degree that nearly vanishes to order at least on . Then .

Proof: Suppose for contradiction that . Then from (6) we have

for ; thus there is a non-trivial linear dependence between the (nonstandard) vectors for . But, from the formula for the Vandermonde determinant, this would imply that two of the are equal, which is absurd.

In the converse direction, we can obtain polynomials that vanish to a high order on , but with degree larger than :

Proposition 11 (Upper bound) Let be positive quantities of polylogarithmic size such that

Then there exists a non-trivial good polynomial of degree at most that vanishes to order on . Furthermore, has height at most

Proof: We use the pigeonholing argument of Thue and Siegel. Let be an positive quantity of quasipolynomial size to be chosen later, and choose coefficients for that are nonstandard natural numbers between and . There are possible ways to make such a selection. For each such selection, we consider the expressions arising as the left-hand side of (6) with and . These expressions are nonstandard integers whose magnitude is bounded by

which by asymptotic calculus simplifies to be bounded by

The number of possible values of these expressions is thus

By the hypothesis and asymptotic calculus, we can make this quantity less than for some of size

In particular, can be taken to be of polylogarithmic size. Thus, by the pigeonhole principle, one can find two choices for the coefficients which give equal values for the expressions in the left-hand side of (6). Subtracting those two choices we obtain the result.

— 3. The bootstrap —

At present, there is no contradiction between the lower bound in Proposition 10 and the upper bound in Proposition 11, because there is plenty of room between the two bounds. To bridge the gap between the bounds, we need a bootstrap argument that uses vanishing on one to imply vanishing (to slightly lower order) on a larger . The key bootstrap in this regard is:

Proposition 12 (Bootstrap) Let be unbounded polylogarithmic quantities, such that

Let be a good polynomial of degree at most and height , that nearly vanishes to order on . Then also vanishes to order on for any .

Proof: It is convenient to use complex analysis methods. We consider the entire function

thus by (3)

By hypothesis, we have

for all and . We wish to show that

for and . Clearly we may assume that .

Fix and . To estimate , we consider the contour integral

(oriented anticlockwise), where is to be chosen later, and estimate it in two different ways. Firstly, we have

so for , we have the bound

when , which by the hypotheses and asymptotic calculus, simplifies to

Also, when we have

We conclude the upper bound

for the magnitude of (7). On the other hand, we can evaluate (7) using the residue theorem. The integrand has poles at and at . The simple pole at has residue

Now we consider the poles at . For each such , we see that the first derivatives of are quasiexponentially small at . Thus, by Taylor expansion (and asymptotic calculus), one can express as the sum of a polynomial of degree with quasiexponentially small coefficients, plus an entire function that vanishes to order at . The latter term contributes nothing to the residue at , while from the Cauchy integral formula (applied, for instance, to a circle of radius around ) and asymptotic calculus, we see that the former term contributes a residue is quasiexponentially small. In particular, it is less than . We conclude that

We have

and thus

choosing to be a large standard multiple of and using the hypothesis , we can simplify this to

To improve this bound, we use the integrality gap. Recall that from (5) that

in particular, is quasiexponentially close to a (nonstandard) integer. Since

we have

(say). Using the integrality gap, we conclude that

which implies that nearly vanishes to order on , as required.

Now we can finish the proof of Proposition 7 (and hence Proposition 6). We select quantities of polylogarithmic size obeying the bounds

and

with a gap of a positive power of between each such inequality. For instance, one could take

many other choices are possible (and one can optimise these choices eventually to get a good value of exponent in Proposition 6).

Using Proposition 11, we can find a good polynomial which vanishes to order on , of height , and hence (by the assumptions on ) of height .

Applying Proposition 12, nearly vanishes to order on for any . Iterating this, an easy induction shows that for any standard , nearly vanishes to order on for any . As was chosen to be larger than a positive power of , we conclude that nearly vanishes to order at least on for any of polylogarithmic size. But for large enough, this contradicts Proposition 10.

Remark 1 The above argument places a lower bound on quantities such as

for integer . Baker’s theorem, in its full generality, gives a lower bound on quantities such as

for algebraic numbers , which is polynomial in the height of the quantities involved, assuming of course that are multiplicatively independent, and that all quantities are of bounded degree. The proof is more intricate than the one given above, but follows a broadly similar strategy, and the constants are completely effective.