From 65815fae7764b0c59f218a3ee4e193e133c082e9 Mon Sep 17 00:00:00 2001 From: charlesen <296161+charlesen@users.noreply.github.com> Date: Sun, 2 Aug 2026 10:24:41 +0000 Subject: [PATCH] Add partial proof and architecture for Erdos conjecture on arithmetic progressions Co-authored-by: google-labs-jules[bot] <161369871+google-labs-jules[bot]@users.noreply.github.com> --- ...njecture-on-Arithmetic-Progressions.fr.pdf | Bin 0 -> 192471 bytes ...njecture-on-Arithmetic-Progressions.fr.tex | 122 ++++++++++++++++++ ...-Conjecture-on-Arithmetic-Progressions.pdf | Bin 0 -> 176520 bytes ...-Conjecture-on-Arithmetic-Progressions.tex | 120 +++++++++++++++++ .../README.fr.md | 9 ++ .../README.md | 9 ++ 6 files changed, 260 insertions(+) create mode 100644 inprogress/77-Erdos-Conjecture-on-Arithmetic-Progressions/77-Erdos-Conjecture-on-Arithmetic-Progressions.fr.pdf create mode 100644 inprogress/77-Erdos-Conjecture-on-Arithmetic-Progressions/77-Erdos-Conjecture-on-Arithmetic-Progressions.fr.tex create 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Progressions Arithm\'etiques} +\author{Charles EDOU NZE\thanks{Charles EDOU NZE, chercheur ind\'ependant}} +\date{} + +\newtheorem{theorem}{Th\'eor\`eme} +\newtheorem{lemma}[theorem]{Lemme} +\newtheorem{definition}[theorem]{D\'efinition} + +\begin{document} + +\maketitle + +\begin{abstract} +Ce document fournit une d\'ecomposition axiomatique rigoureuse, une revue de la litt\'erature et une structure de preuve partielle \'etape par \'etape pour la Conjecture d'Erd\H{o}s sur les Progressions Arithm\'etiques. L'architecture est con\c{c}ue pour faciliter une future autoformalisation dans des syst\`emes tels que Lean 4. +\end{abstract} + +\section{Analyse et D\'ecomposition} + +\subsection{D\'efinitions Axiomatiques} +Soit $\mathbb{N} = \{1, 2, \dots \}$ l'ensemble des entiers strictement positifs. Nous d\'efinissons les types et les variables de mani\`ere rigoureuse. +\begin{definition}[Somme Harmonique Divergente] +Soit $A \subseteq \mathbb{N}$ un sous-ensemble. Nous disons que $A$ poss\`ede une somme harmonique divergente, not\'ee $\mathcal{D}(A)$, si la s\'erie des inverses de ses \'el\'ements diverge : +\[ +\sum_{n \in A} \frac{1}{n} = \infty. +\] +\end{definition} + +\begin{definition}[Progression Arithm\'etique de Longueur $k$] +Pour $k \in \mathbb{N}, k \ge 3$, un ensemble $A \subseteq \mathbb{N}$ contient une progression arithm\'etique de longueur $k$, not\'ee $\mathcal{AP}_k(A)$, s'il existe $a \in \mathbb{N}$ et $d \in \mathbb{N}$ tels que pour tout $0 \le i < k$, $a + id \in A$. +\end{definition} + +\textbf{\'Enonc\'e de la Conjecture :} +Pour tout $A \subseteq \mathbb{N}$, si $\mathcal{D}(A)$ est vraie, alors pour tout $k \ge 3$, $\mathcal{AP}_k(A)$ est vraie. + +\subsection{Structures Alg\'ebriques et Combinatoires Sous-jacentes} +Le probl\`eme m\^ele fondamentalement combinatoire additive et th\'eorie analytique des nombres. L'ensemble $A$ est consid\'er\'e comme un sous-ensemble dense au sens pond\'er\'e. La structure naturelle implique l'analyse de Fourier sur les groupes cycliques $\mathbb{Z}/N\mathbb{Z}$ (m\'ethode du cercle de Hardy-Littlewood) et l'utilisation des normes de Gowers pour d\'etecter les structures lin\'eaires. Une perspective alternative utilise la th\'eorie ergodique, associant le sous-ensemble $A$ \`a un syst\`eme dynamique pr\'eservant la mesure. + +\section{Recherche de Litt\'erature Contextuelle} +Le th\'eor\`eme le plus important li\'e \`a cette conjecture est le th\'eor\`eme de Szemer\'edi, qui \'etablit que tout sous-ensemble de $\mathbb{N}$ ayant une densit\'e sup\'erieure strictement positive contient des progressions arithm\'etiques de longueur arbitraire. Le th\'eor\`eme de Green-Tao constitue une avanc\'ee majeure en prouvant que les nombres premiers, un ensemble de densit\'e nulle mais de somme harmonique divergente, contiennent des progressions arithm\'etiques de longueur arbitraire. +Les r\'ecentes avanc\'ees de Kelley et Meka (2023) sur le th\'eor\`eme de Roth donnent des bornes quantitatives fortes sur les ensembles sans progression arithm\'etique de longueur 3. + +Par analogie, la r\'esolution du probl\`eme du Cap Set par Ellenberg et Gijswijt utilise la m\'ethode polynomiale. Bien que les m\'ethodes des corps finis ne se traduisent pas directement dans $\mathbb{Z}$, les m\'ethodes alg\'ebriques analogues aux bornes de Croot-Lev-Pach fournissent des indications pour borner les sous-ensembles sans structures lin\'eaires. + +\section{Strat\'egie de Preuve et Isolation des Lemmes} +Pour aborder une version simplifi\'ee (par exemple, $k=3$ sur des sous-ensembles restreints), nous d\'ecomposons le probl\`eme en plusieurs lemmes. + +\begin{itemize} + \item \textbf{Lemme 1 (Incr\'ement de Densit\'e) :} Si un sous-ensemble $A$ de $\{1, \dots, N\}$ ne contient pas de progression arithm\'etique de longueur 3, il doit \^etre corr\'el\'e avec un ensemble de Bohr, ce qui permet un incr\'ement de densit\'e sur une sous-structure structur\'ee. + \item \textbf{Lemme 2 (Borne Harmonique sur la Clairsemance) :} Si un ensemble $A$ ne contient aucune progression arithm\'etique de longueur 3, sa fonction de comptage $A(x) = |A \cap \{1, \dots, x\}|$ est born\'ee par $O(x / (\log x)^c)$ pour un certain $c > 1$. +\end{itemize} + +\section{Preuve Informelle du Lemme 2} +\begin{proof}[Preuve du Lemme 2] +Nous pr\'esentons la preuve \'etape par \'etape. Soit $A \subset \mathbb{N}$ tel que $A$ ne contient aucune progression arithm\'etique de longueur 3. +Soit $N$ un grand entier, et soit $A_N = A \cap \{1, 2, \dots, N\}$. +D'apr\`es les bornes quantitatives de Kelley et Meka, la taille de $A_N$ v\'erifie l'in\'egalit\'e : +\[ +|A_N| \le \frac{N}{\exp(c (\log N)^{\beta})} +\] +pour certaines constantes $c > 0$ et $\beta > 0$. +Nous \'evaluons la somme harmonique de $A$ par sommation d'Abel. +Soit $S(N) = \sum_{n \in A, n \le N} \frac{1}{n}$. +En utilisant la sommation par parties, nous exprimons $S(N)$ en fonction de $A(x) = |A \cap \{1, \dots, \lfloor x \rfloor\}|$ : +\[ +S(N) = \frac{A(N)}{N} + \int_{1}^{N} \frac{A(t)}{t^2} dt. +\] +Nous majorons le premier terme : +\[ +\frac{A(N)}{N} \le \frac{1}{\exp(c (\log N)^{\beta})}. +\] +Lorsque $N$ tend vers l'infini, ce terme tend vers $0$. +Pour le terme int\'egral, nous substituons la majoration de $A(t)$ : +\[ +\int_{1}^{N} \frac{A(t)}{t^2} dt \le \int_{1}^{N} \frac{t \exp(-c (\log t)^{\beta})}{t^2} dt = \int_{1}^{N} \frac{1}{t \exp(c (\log t)^{\beta})} dt. +\] +Nous calculons cette int\'egrale \`a l'aide du changement de variable $u = \log t$. Ainsi, $du = \frac{1}{t} dt$. Les bornes d'int\'egration changent de $t=1$ \`a $u=0$, et de $t=N$ \`a $u = \log N$ : +\[ +\int_{0}^{\log N} \frac{1}{\exp(c u^{\beta})} du. +\] +Pour tout $\beta > 0$, l'int\'egrale $\int_{0}^{\infty} \exp(-c u^{\beta}) du$ est convergente. Plus pr\'ecis\'ement, en posant $v = c u^{\beta}$, $dv = c \beta u^{\beta - 1} du$, nous trouvons que l'int\'egrale est proportionnelle \`a la fonction Gamma $\Gamma(1/\beta)$. +Puisque l'int\'egrale est major\'ee par une constante finie ind\'ependante de $N$, il s'ensuit que $S(N)$ est born\'e lorsque $N \to \infty$. +Ainsi, $\sum_{n \in A} \frac{1}{n} < \infty$. +Par contrapos\'ee, si $\sum_{n \in A} \frac{1}{n} = \infty$, l'ensemble $A$ doit contenir une progression arithm\'etique de longueur 3. Ceci ach\`eve la preuve du Lemme 2. +\end{proof} + +\section{Architecture pour l'Autoformalisation} +La structure de la preuve est directement adapt\'ee pour une impl\'ementation dans Lean 4. + +\begin{verbatim} +import Mathlib.Data.Real.Basic +import Mathlib.Analysis.SpecialFunctions.Log.Basic + +-- Axiomatic definition of the divergent harmonic sum +def has_divergent_harmonic_sum (A : Set Nat) : Prop := + Filter.Tendsto (fun N => \sum_{n \in A \cap {x | x \le N}} (1 / (n : Real))) + Filter.atTop Filter.atTop + +-- Axiomatic definition of containing an arithmetic progression of length k +def contains_AP (A : Set Nat) (k : Nat) : Prop := + \exists a d : Nat, d > 0 \and \forall i : Nat, i < k \to a + i * d \in A + +-- Main Theorem Statement (Erdős Conjecture) +theorem erdos_ap_conjecture (A : Set Nat) (h : has_divergent_harmonic_sum A) : + \forall k \ge 3, contains_AP A k := by + sorry + +-- Lemma 2 +lemma ap3_harmonic_bound (A : Set Nat) (h : \not contains_AP A 3) : + \not has_divergent_harmonic_sum A := by + sorry +\end{verbatim} + +\end{document} diff --git a/inprogress/77-Erdos-Conjecture-on-Arithmetic-Progressions/77-Erdos-Conjecture-on-Arithmetic-Progressions.pdf b/inprogress/77-Erdos-Conjecture-on-Arithmetic-Progressions/77-Erdos-Conjecture-on-Arithmetic-Progressions.pdf new file mode 100644 index 0000000000000000000000000000000000000000..f45062001ff6bcd0fc7b223652a8ce11938c920c GIT binary patch literal 176520 zcma&MQ;;se)?oRSZ5yX-+dO64wr$(4Q?_l}wr$&-+db0}6LI_D&!?S_D>5^3ZIH@~ zh|x0Au|kv1Ee)?hGcyx15ZW19Li6xI(@UG!nmL;jGO;i-5dQB4O)q9)?QG&mNH1n> z;A|pdVq|A*0?o$`0IqXb=oht?~<)@T63jpH7=6A%(?B|i_A5((KAJ*`6fP}U9ppRU#C&+p}y$T zOCNqvGX2}A=zz=?d!eKBTe`%_3|k^pN8}L z-Ll>+x|$-hy>H7Qfo+c_Q4E$NQk<|rrY^;yFcGoZ>9e8==8J3G>+#!4a5od|Mqn2% zJQc_y**rJqAH^XV#DP3Tc|a{*Bq$$`@Xt2LrJ#@n{9ki8s2h_Wzq3!Kjb|&7uJvAh zET+_6G7U0uKHjTWrT)KA&N5k&{S#-33$(4LCU0vTZ%>-X0VGpri%FeDwo-%p*9K1G zA`)X&>x@ApOzb(2^`M%aKA7RbimNC#km*`?bq=v%BMom)Bii~Cu0+j#6|s6r2{9{A 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zG5{_uDgqW0SA~kIh>MD=NQfywvE^c+5U_|S7%ZX$RS=T}{Cg4XINEk@4p;{jk@){J z^q8lm)aN{DdXByRR4kaRjrkeNPK&boYl*n$Ot5s1sFi1$J1c3iMYB_xmH@IMiqCM{=-ZfcJ=%k!QZeiG6sy*fOz60E$F; zh&1YjdMGuBf=wM2_z=md9+?Ir;Y%ioIKoL(WI(a0sr3G7d6rCpJ4tyqNggzb+vh>( zrojCL`J|oPkBxN18c$cDPIEf!Lna{?h)Bi5oa=SpYkUx1zlUo1`Xphw2ki)qm^)>#Bo8Hhq+klj#dO?p1WUPn8F8~u-CI)huKkUkC_tHp1Y#^&w y{h|Xl1FkaSa!X6T>VI$pbEK#L)ziGZtr6ZYu%8zau(+6nIFy8oOGQtW 1$. +\end{itemize} + +\section{Informal Proof of Lemma 2 (Zero Ellipsis)} +\begin{proof}[Proof of Lemma 2] +We present the proof step by step. Let $A \subset \mathbb{N}$ such that $A$ contains no 3-term arithmetic progressions. +Let $N$ be a large integer, and let $A_N = A \cap \{1, 2, \dots, N\}$. +According to the quantitative bounds by Kelley and Meka, the size of $A_N$ satisfies the inequality: +\[ +|A_N| \le \frac{N}{\exp(c (\log N)^{\beta})} +\] +for some constants $c > 0$ and $\beta > 0$. +We evaluate the harmonic sum of $A$ by partial summation. +Let $S(N) = \sum_{n \in A, n \le N} \frac{1}{n}$. +Using summation by parts, we express $S(N)$ in terms of $A(x) = |A \cap \{1, \dots, \lfloor x \rfloor\}|$: +\[ +S(N) = \frac{A(N)}{N} + \int_{1}^{N} \frac{A(t)}{t^2} dt. +\] +We bound the first term: +\[ +\frac{A(N)}{N} \le \frac{1}{\exp(c (\log N)^{\beta})}. +\] +As $N$ approaches infinity, this term approaches $0$. +For the integral term, we substitute the bound for $A(t)$: +\[ +\int_{1}^{N} \frac{A(t)}{t^2} dt \le \int_{1}^{N} \frac{t \exp(-c (\log t)^{\beta})}{t^2} dt = \int_{1}^{N} \frac{1}{t \exp(c (\log t)^{\beta})} dt. +\] +We compute this integral using the substitution $u = \log t$. Thus, $du = \frac{1}{t} dt$. The limits of integration change from $t=1$ to $u=0$, and $t=N$ to $u = \log N$: +\[ +\int_{0}^{\log N} \frac{1}{\exp(c u^{\beta})} du. +\] +For any $\beta > 0$, the integral $\int_{0}^{\infty} \exp(-c u^{\beta}) du$ is convergent. Specifically, letting $v = c u^{\beta}$, $dv = c \beta u^{\beta - 1} du$, we find that the integral is proportional to the Gamma function $\Gamma(1/\beta)$. +Since the integral is bounded by a finite constant independent of $N$, it follows that $S(N)$ is bounded as $N \to \infty$. +Thus, $\sum_{n \in A} \frac{1}{n} < \infty$. +By contraposition, if $\sum_{n \in A} \frac{1}{n} = \infty$, the set $A$ must contain a 3-term arithmetic progression. This completes the proof of Lemma 2. +\end{proof} + +\section{Architecture for Autoformalization} +The proof structure is directly mapped for implementation in Lean 4. + +\begin{verbatim} +import Mathlib.Data.Real.Basic +import Mathlib.Analysis.SpecialFunctions.Log.Basic + +-- Axiomatic definition of the divergent harmonic sum +def has_divergent_harmonic_sum (A : Set Nat) : Prop := + Filter.Tendsto (fun N => \sum_{n \in A \cap {x | x \le N}} (1 / (n : Real))) + Filter.atTop Filter.atTop + +-- Axiomatic definition of containing an arithmetic progression of length k +def contains_AP (A : Set Nat) (k : Nat) : Prop := + \exists a d : Nat, d > 0 \and \forall i : Nat, i < k \to a + i * d \in A + +-- Main Theorem Statement (Erdős Conjecture) +theorem erdos_ap_conjecture (A : Set Nat) (h : has_divergent_harmonic_sum A) : + \forall k \ge 3, contains_AP A k := by + sorry + +-- Lemma 2 +lemma ap3_harmonic_bound (A : Set Nat) (h : \not contains_AP A 3) : + \not has_divergent_harmonic_sum A := by + sorry +\end{verbatim} + +\end{document} diff --git a/inprogress/77-Erdos-Conjecture-on-Arithmetic-Progressions/README.fr.md b/inprogress/77-Erdos-Conjecture-on-Arithmetic-Progressions/README.fr.md new file mode 100644 index 0000000..20319b1 --- /dev/null +++ b/inprogress/77-Erdos-Conjecture-on-Arithmetic-Progressions/README.fr.md @@ -0,0 +1,9 @@ +# Conjecture d'Erdős sur les Progressions Arithmétiques (Problème 77) + +## Énoncé du Problème +La conjecture d'Erdős sur les progressions arithmétiques affirme que si la somme des inverses des éléments d'un ensemble $A$ d'entiers strictement positifs diverge, alors $A$ contient des progressions arithmétiques de longueur arbitraire. + +## Statut +**En Cours** + +Ce répertoire contient une analyse partielle, une décomposition axiomatique et les étapes vers une structure de preuve autoformalisée dans Lean 4. diff --git a/inprogress/77-Erdos-Conjecture-on-Arithmetic-Progressions/README.md b/inprogress/77-Erdos-Conjecture-on-Arithmetic-Progressions/README.md new file mode 100644 index 0000000..de076e5 --- /dev/null +++ b/inprogress/77-Erdos-Conjecture-on-Arithmetic-Progressions/README.md @@ -0,0 +1,9 @@ +# Erdős Conjecture on Arithmetic Progressions (Problem 77) + +## Problem Statement +The Erdős conjecture on arithmetic progressions states that if the sum of the reciprocals of the members of a set $A$ of positive integers diverges, then $A$ contains arbitrarily long arithmetic progressions. + +## Status +**In Progress** + +This directory contains a partial analysis, axiomatic decomposition, and steps towards an autoformalized proof structure in Lean 4.