14
$\begingroup$

For any set $B\subseteq \mathbb{N}$ one can associate the formal series $$f_B(z) = \sum_{b\in B}z^b$$ and obtain $$f_B(z)^k = \sum_{n\geqslant 0} r_{B,k}(n)z^n,$$ where $r_{B,k}(n) = |\{(x_1,\cdots,x_k)\in B^k : x_1+\cdots+x_k=n\}|$. As it is characteristic of the Circle Method, one them have $$ r_{B,k}(n) = \frac{1}{2\pi i}\int_{|z|=\rho} \frac{f_B(z)^k}{z^{n+1}}~\mathrm{d}z = \int_0^1 f_B(e(t))^k e(-nt)~\mathrm{d}t,$$ where $e(z) = e^{2\pi iz}$.

I am starting to study the Circle Method and the first thing I noticed is that, although it is a very powerful way to deal with both Waring's and weak Goldbach's problems, it seems that it relies heavily on the number-theoretical properties of the primes and Waring bases to work properly, and doesn't have strong "combinatorial powers" per se.

However, this first step of the setup seems to be a very very natural starting point to deal with problems involving (asymptotic) bases in general, but I found way less discussions about this in the literature than I expected. More especifically, my question is:

Is there any approach of the Erdős–Turán conjecture on additive bases via the Hardy-Littlewood circle method? Or it is known to be an ineffective approach?

As far as I know, the best results up to date has been made in the closely related problem of finding thin bases, namely

  • Erdös-Tetali theorem (1990) on the existence of asymptotic $h$-bases $B$ with $r_{B,h}(n) = \Theta(\log(n))$, using, basically, only probabilistic methods.

  • Vu's thin subbases of Waring bases (2000), showing the existence of asymptotic $h_k$-bases $B^{(k)} \subseteq P_k = \{n^k : n\geqslant 0\}$ with $r_{B^{(k)},h_k}(n) = \Theta(\log(n))$ for every $k$. He uses both probabilistic methods AND the Circle Method, but as far as I understood the last one is used "only" in dealing with the $P_{k}$'s. (By "only" I mean that it is not in the sense that I am asking in this question.)

Other approaches like the Borwein-Choi-Chu result (2006) that asserts that if $B$ is a $2$-basis then $\liminf\limits_{n\to\infty} r_{B,2}(n) > 7$ seems fruitful too, but I didn't spent enough time reading it to be able to make a proper remark. On the other hand, the current probabilistic methods apparently have a "technical limitation" involving the $\Theta(\log(n))$, as it seems to be the best one can get in this direction.

After reading Xiao's master's dissertation and a few books (Alon & Spencer's Probabilistic Method, Nathanson's Additive NT: Classical Bases, Tao & Vu's Additive Combinatorics), I had a feeling that the Circle Method is a possible way to understand better this limitation of the probabilistic method.

In any case, this is more a raw idea than anything. I'd appreciate any references, thoughts and thinghs to think about. Thanks in advance!

$\endgroup$
6
$\begingroup$

Your question probably doesn't have a definitive answer, as it is unlikely that one can prove that the `circle method' (it has evolved in many directions taking in tools from a very broad area of mathematics, so it is likely difficult to define exactly what the circle method entails) can never be used to prove the Erdös-Turán conjecture for additive bases (colloquially, the 'Erdös-Turán conjecture' is usually reserved for this: https://en.wikipedia.org/wiki/Erd%C5%91s_conjecture_on_arithmetic_progressions). That said, I would argue that the current knowledge of the circle method is unlikely to produce significant results on the conjecture.

The reason is that we expect a basis $B$ such that $r_{B,k}(n)$ has as small an order of magnitude (which is conjectured by Erdös to be $\log n$) as possible is very close to the random sets constructed by Erdös and Erdös-Tetali. To date, we have yet to produce a single concrete example of such a set, so it is unclear what properties, if any, such a set should have. However, as you mentioned the circle method is heavily dependent on certain key ingredients to work. Indeed, it mostly works in settings somewhat related to algebraic varieties because then you can expect that your exponential sums turn into Weyl sums, where you can obtain savings via Weyl differencing (this explanation is due to Sam Chow). Even then, the circle method seems to be fairly inefficient in the sense that many more variables are needed than expected to produce acceptable savings.

One way to think about the weakness in the circle method to tackle problems of this type is in the proof of Vinogradov's theorem. The set of primes are in some sense extremely ripe to be a candidate for an additive basis: for every admissible congruence class they are equidistributed, they are very dense (has density exceeding $\Omega(n^{1-\epsilon})$ for any $\epsilon > 0$), and there are many heuristics which point to them being `random'. Even for the set of primes, the circle method can only succeed in proving that they are an additive basis of order 4 (i.e. every (sufficiently large) positive integer is the sum of four primes, a fact implied by Vinogradov's theorem which asserts that every sufficiently large odd number is the sum of three primes. Recent work of Helfgott seems to have removed the 'sufficiently large' requirement, though I believe this work is still being refereed), while the expected truth is that they are an additive basis of order 3 (Goldbach's conjecture). If you examine why the circle method has (to date) failed to produce a proof of the Goldbach conjecture, I think you will gain a better understanding as to why the circle method likely cannot be applied to a set whose only known property is 'thin additive basis'.

To further clarify, my point is the following: the additive bases which are as sparse as possible are much sparser than the primes, so if the circle method has failed to prove the Goldbach conjecture, then it is unlikely that it can produce results for a set which is in almost all ways much harder to work with (i.e. has less arithmetical structure and much sparser).

$\endgroup$
  • 2
    $\begingroup$ The circle method has failed to prove the (tenary) Goldbach conjecture? I think you'd be more updated on that... $\endgroup$ – Fan Zheng Oct 3 '15 at 23:58
  • 3
    $\begingroup$ The question was phrased in terms of additive bases. The ternary Goldbach conjecture was established by IM Vinogradov in 1937, which asserts that every sufficiently large odd positive integer is the sum of three primes. This in particular shows that every sufficiently large positive integer is the sum of at most four primes. The Goldbach problem asserts that every positive integer larger than 2 is the sum of two primes, which implies that every positive integer at least 5 is the sum of at most three primes. $\endgroup$ – Stanley Yao Xiao Oct 4 '15 at 0:14
  • 2
    $\begingroup$ Haven't you heard about all the fuss about Helfgott? $\endgroup$ – Fan Zheng Oct 4 '15 at 0:32
  • 4
    $\begingroup$ I think you misunderstand what I posted. I did not make a distinction between the ternary Goldbach conjecture and the asymptotic ternary Goldbach conjecture (this is the convention I took in my masters' thesis). I am only concerned with the statement that all sufficiently large odd numbers is the sum of three primes. Helfgott's work is certainly important, but I did not feel it had much to do with what I wanted to say. Apparently the main point I was trying to make in the answer did not come across clearly, so I will edit to clarify my intent. $\endgroup$ – Stanley Yao Xiao Oct 4 '15 at 0:42
  • 1
    $\begingroup$ @ChrisTáfula: Sam and I had a discussion about this question on Facebook, and that was the remark he made. I am not sure if he made this particular remark in any of his publications. You can see his existing publications and his contact info here: www2.maths.bris.ac.uk/~sc12537 $\endgroup$ – Stanley Yao Xiao Oct 4 '15 at 17:42

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.