nt.number theory – When did the Main Conjecture in Vinogradov’s Mean Value Theorem first appear in literature?

Recently I was asked about the history of Vinogradov’s Mean Value Theorem that I was hoping someone here could clarify. Let me first start with some terminology. Let $J_{s, k}(X)$ be the number of $2s$-tuples $(x_1, ldots, x_s, y_1, ldots, y_s)$ such that
begin{align*}
x_1 + cdots + x_{s} &= y_{1} + cdots + y_{s}\
x_{1}^{2} + cdots + x_{s}^{2} &= y_{1}^{2} + cdots + y_{s}^{2}\
&vdots\
x_{1}^{k} + cdots + x_{s}^{k} &= y_{1}^{k} + cdots + y_{s}^{k}
end{align*}

for $1 leq x_{i}, y_{i} leq X$. It is not hard to see that $$J_{s, k}(X) gtrsim_{s, k} X^{s} + X^{2s – frac{1}{2}k(k + 1)}.$$

The now proven Main Conjecture in Vinogradov’s Mean Value Theorem is that this lower bound is essentially an upper bound. More precisely, the conjecture was:

Conjecture: For every $epsilon > 0$, $$J_{s, k}(X) lesssim_{epsilon, s, k} X^{epsilon}(X^{s} + X^{2s – frac{1}{2}k(k + 1)}).$$

This conjecture follows from classical methods for $k = 2$, first proven by Wooley for $k = 3$ using efficient congruencing in 2014 and then proven by Bourgain, Demeter, and Guth for $k geq 4$ using decoupling methods in 2015.

My question is: when did this conjecture as stated above first appear in the literature?

Looking through Vinogradov’s 1935 paper “New estimates for Weyl sums”, it seems that this conjecture is not stated. The term “Vinogradov’s Mean Value Theorem” referring to any bound of the form $J_{s, k}(X)lesssim X^{2s – frac{1}{2}k(k + 1) + Delta_{s, k}}$ for some $Delta_{s, k}$ positive and $s gtrsim k^{2}log k$ seems to appear in print as early as 1947 or 1948 in these two works by Hua:

  1. Page 49 of the Russian version of his Additive Theory of Prime Numbers (http://mi.mathnet.ru/eng/tm1019)
  2. In Hua’s paper “An Improvement of Vinogradov’s Mean-Value Theorem and Several Applications” (https://doi.org/10.1093/qmath/os-20.1.48)

Though in both, it seems to imply that this term was in use as early as 1940. However neither also state the conjecture as mentioned above.

co.combinatorics – A conjecture on circular permutations of n elements in an abelian group of odd order

In 2013 I formulated the following conjecture in additive combinatorics.

Conjecture. Let $G$ be an additive abelian group of odd order, and let $A$ be a subset of $G$ with $|A|=n>2$. Then, there is a circular permutation $(a_1,ldots,a_n)$ of all the elements of $A$ such that all the adjacent sums $ a_1+a_2,ldots,a_{n-1}+a_n,a_n+a_1$
are pairwise distinct.

Recently, Mr. Yu-Xuan Ji, a student at Nanjing Univ., verified this conjecture for $|G|<30$.

I’m even unable to show the conjecture for $G=mathbb Z/pmathbb Z$ with $p$ an odd prime.

Any ideas towards the solution of this conjecture? Your comments are welcome!

algebraic curves – The proof of Toeplitz’ conjecture fail

There is an unsolved problems in mathematics call inscribed square problem or Toeplitz’ conjecture.

Does every Jordan curve admit an inscribed square?

Based on the Lemma 1 as follows I give a counter example of the Toeplitz’ conjecture.

Lemma 1: Let $ABCD$ and $A’B’C’D’$ are two square then four lines $AA’, BB’, CC’, DD’$ form a orthodiagonal quadrilateral. See the Figure as folows:

enter image description here

My question 2: Could You send my the reference for the Lemma 1?

Since Lemma 1, We deduce that: If $ABCD$ is not orthodiagonal quadrilateral then exist maximum one square incribed the quadrilateral $ABCD$ which one vertex of the square lie on one segment $AB$, $BC$, $CD$, $DA$ respectively.

Now I give an example:

Take a square with vertex $A(0,0)$, $B(0,1)$, $C(1,1)$, $D(0,1)$ so $ABCD$ is a square. Let $E(2,-5)$, $F(3,4)$. Now we consider a convex quadrilateral form by lines $EA$, $EB$, $FC$, $FD$. Detail: $FD cap EB=G$, $FD cap EA=H$, $FC cap EA=I$, $FC cap EB=J$. This is the quadrilateral $GHIJ$ also call curve $GHIJ$ (see Figure below).

enter image description here

We have $ABCD$ is a square and $GI$ is not perpendicular to $HJ$ by the Lemma 1 we have no exist another square such that the one vertices of the square lie on one lines $GH$, $HI$, $IJ$, $JG$ respectively. But $B, C$ don’t lie on any segment $GH$, $HI$, $IJ$, $JG$. So square $ABCD$ is not inscribed the curve $GHIJ$.

So now we find a square such that two vertices lie on the same segments $GH$, $HI$, $IJ$ or $JG$ and two vertices lie on other segment. But by the construction in Inner Inscribed Squares Triangle, Outer Inscribed Squares Triangle, the answer is no exist any square such that two vertices lie on the same segment and two other vertices lie on other segment.

How to Prove this conjecture?

I am starting to work on developing a generalization of the Hodge Index Theorem, and I recently came up with the following conjecture, the proof of which I am struggling to make progress on:

Conjecture. Let X be a noetherian scheme, and let $mathcal{L}$ be an invertible sheaf on X. Let $d = dim X$, and let $q = d – i$ for $1 < i leq d$. We associate to every cycle $Z in mathcal{Z}^{i}(X)$ a subcycle $N(Z) subseteq Z$, defined to be the subset of $Z$ numerically equivalent to some integer $n$. Then for every $Z in mathcal{Z}^{i}(X)$, $mathcal{L}(Z)$ is ample if for any two nonempty subsets $P, Q subset Z$ (including the case $P = Q$), the following two conditions are satisfied:

  1. $|mathrm{Tor}_{mathcal{L}(Z)}^{n}(P,Q)| geq |bigwedge^{n}(N(Z))|$;
  2. $mathcal{L}(Z) cong komega_{Z}$ for some $k in mathbb{Z}$.

How should I approach this?

Connes embedding conjecture

In the blog of Scott Aaronson,https://www.scottaaronson.com/blog/?p=4512 he mentioned that the Connes embedding conjecture is false. I wonder whether the results in their paper are correct or not.

Try to find a weaker ABC conjecture but that is usefull as the ABC conjecture

My question: The conjecture as follows is weaker than the ABC conjecture but that is usefull as the ABC conjecture?

Cited at here

Given a positive integer $P>1$, let its prime factorization be written

$$P=p_1^{a_1}p_2^{a_2}p_3^{a_3}…p_k^{a_k}$$

Define the functions $h(P)$, $d(P)$ and $rad'(P)$ by $h(1)=1$ and
$h(P)=min(a_1, a_2,..,a_k)$

Let $g=gcd(a_1, a_2,…, a_k)$

$$d(P)=frac{h(P)}{g}=min(frac{a_1}{g},frac{a_2}{g},…,frac{a_k}{g})$$
$$rad'(P)=(p_1p_2…p_k)^{d(P)}$$ Some examples: 1) Let
$P=2^5.5^7.11^8$ then $rad(P)=2.5.11$ and $rad'(P)=2^5.5^5.11^5$ 2)
$P=17^8$ then $rad(P)=17$ and $rad'(P)=17$
There are some simple properties of $rad'(P)$ 1) $rad'(P)=rad(P)^{d(P)}$ 2) $rad(P) le rad'(P) le P$ 3)
$rad'(P^n)=rad'(P) le P$ 4) In general case $rad'(AB) ne
> rad'(A)rad'(B)$

Conjecture1: For every positive real number $varepsilon >0$, the inequality $$A+B > (rad'(A).rad'(B).rad'(A+B))^{1+varepsilon}$$ has
only finitely relatively prime integers $A$ and $B$.

Let $varepsilon=0$ here are some examples $rad(AB(A+B)) < A+B<
rad'(A).rad'(B).rad'(A+B)$

enter image description here

nt.number theory – Status of the $n$ conjecture, and what about ideas or techniques that could be potentially interesting as a reference request

Wikipedia has an article for the known as n conjecture that is a generalization of the abc conjecture. I would like to know what about the current status of the $n$ conjecture, I mean if you as professional mathematician, or your colleagues (professors), can tell something new about it, about the status of this conjecture. I am going to try to follow the references or explanation from your knowledges, I think that it will be interesting for your colleagues of this MathOverflow.

Question 1. Can you tell us what about the current status of the $n$ conjecture, we consider $n>3$, of the $n$ conjecture? Many thanks.

On the other hand I know from an informative point of view an article, (2), where is explained the transfer method, I hope to refer it in the right way, a method to create new abc triples (for a suitable formulation of the abc conjecture) from a given abc triple. I would like to know if there is in the literature a transfer method for some formulation of the n conjecture, here again we are considering $n>3$, alternatively I add other question to bring here more chances to know references for different methods or ideas.

Question 2. Is it known a tranfer method similar than the transfer method for abc triples for any formulation of the $n$ conjecture? In this case refer the literature and I try to search and read what about this method from the literature. Alternatively for this question, please answer as a reference request what can be other theorems, ideas or techniques that can be potentially interesting in the study of the $n$ conjecture (if you need it refer the literature and I try to search and read those from the literature). Many thanks.

In case that there is no such transfer method for the $n$ conjecture with $n>3$, add if you want a discussion about if it is possible/feasible a transfer method for some formulation of the $n$ conjecture (with $n>3$).

References:

(1) Jerzy Browkin and Juliusz Brzeziński, Some remarks on the abc-conjecture, Math. Comp. 62 (206), (1994), pp. 931–939.

(2) Greg Martin and Winnie Miao, abc triples, Functiones et Approximatio Commentarii Mathematici, Funct. Approx. Comment. Math. Vol. 55, N. 2 (2016), pp. 145-176.

nt.number theory – A conjecture involving $P_n=prod_{k=1}^np_k$

For each positive integer $n$ let $P_n=prod_{k=1}^n p_k$, where $p_k$ is the $k$th prime.

Question. Is my following conjecture true?

Conjecture. For any integer $n>1$, there are $k,min{1,ldots,n-1}$ such that
$$P_nequiv P_kpmod n text{and} P_nequiv -P_mpmod n.$$

For example, $P_{32}equiv P_{23}pmod{32}$ and $P_{32}equiv -P_8pmod{32}$.

I have verified the conjecture for all $n=2,3,ldots,70000$. When $n$ is squarefree, the conjecture holds trivially. I’m unable to prove the conjecture fully.

For the motivation of the conjecture, one may look at Conjecture 1.5 and Remark 1.7 in my paper available from http://dx.doi.org/10.1016/j.jnt.2013.02.003.

Your comments are welcome!

nt.number theory – The equivalent proposition of Legendre's conjecture

Legendre's conjecture, proposed by Adrien-Marie Legendre, state that there is a prime number between $ n ^ 2 $ and $ (n + 1) ^ 2 $ for every positive integer $ n $.

My guess: Let $ n $ be a positive integer, $ p_1 $, $ p_2 $ and $ p_3 $ be odd prime numbers, and $ n ge2p_1 $, ($ n ge6 $)

There is at least one prime number $ p_2 $, so that $ n ^ 2 <p_2 le n ^ 2 + n – p_1 $, and there is at least one prime number $ p_3 $, so that $ n ^ 2 + n – p_1 le p_3 <(n + 1) ^ 2 $

ag.algebraic geometry – The flagship conjecture of the theory of geometric complexity

I read the article by Landsberg, which provides an introduction to the theory of geometric complexity. In chapter 2 of this article, the author defined the following objects:

Let $ W = mathbb {C} ^ {n ^ 2} $, $ det_n in S ^ nW $ to be the determining polynomial. And defined
$$ Det_n: = overline {GL (W) cdot (det_n)} subseteq mathbb {P} W $$

Where the bar indicates the closing of Zariski. I am confused with the definition of $ Det_n $. I will use the case $ n = $ 2 to explain my confusion.

When $ n = $ 2, we see $ W = mathbb {C} ^ 4 $ and $ det_2 in S ^ 2W $. Let us note a base of $ W $ to be {$ X_ {11}, X_ {12}, X_ {21}, X_ {22} $}, then $ det_2 = X_ {11} X_ {22} -X_ {12} X_ {21} $. Of course $ GL (W) $ can act on {$ X_ {11}, X_ {12}, X_ {21}, X_ {22} $} and their linear combination, and the result of such an action is always an element $ W $, and therefore an element of $ mathbb {P} V $, but there are product terms in $ det_2 $, so I cannot understand how the result of such an action can be $ mathbb {P} W $.

Thanks for your help!