[Paper Review] Arithmetic progressions in middle Nth cantor sets

Jon Chaika|arXiv (Cornell University)|Mar 27, 2017

Limits and Structures in Graph Theory3 citations

TL;DR

This paper proves that middle $\frac{1}{N}^\text{th}$ Cantor sets contain arithmetic progressions of length proportional to $\frac{N}{\log N}$, using a recursive induction argument on $k$-good intervals to show that intersections of translated copies of the set remain non-empty. The key result establishes the existence of such configurations for all sufficiently large $N$, with a quantitative bound on the length of the progression.

ABSTRACT

We show the middle Nth cantor set contains arithmetic progressions of length at least proportional to N/log_2(N).

Motivation & Objective

• To establish the existence of long arithmetic progressions within middle $\frac{1}{N}^\text{th}$ Cantor sets for large $N$.
• To determine the maximal length of arithmetic progressions that can be embedded in such Cantor sets.
• To develop a recursive method based on interval structure and translation invariance to prove non-emptiness of intersections of translated Cantor sets.
• To analyze the geometric and measure-theoretic constraints imposed by the removal of middle $\frac{1}{N}^\text{th}$ intervals at each stage of the construction.
• To explore the robustness of the result under bilipschitz maps and to assess the tightness of the derived bounds.

Proposed method

• Define $X$ as the middle $\frac{1}{N}^\text{th}$ Cantor set via iterative removal of the middle $\frac{1}{N}^\text{th}$ from each interval in $C_k$.
• Introduce the concept of a $k$-good interval $J$ of size $\frac{1}{N^k}$, meaning it intersects $\frac{N}{2}$ disjoint intervals of size $\frac{1}{N^{k+1}}$ in the translated sets $X_{\frac{1}{N^{k+1}}} + a_i$.
• Use induction on $k$, proving that if $J$ is $k$-good, then it contains a subinterval $J'$ that is $k+1$-good.
• Apply Lemma 4 to bound the number of intervals removed at each stage that can intersect a given interval $J$, showing it is at most $2^{L-k-1}$ for $L > k$.
• Use Corollary 5 to estimate the number of intervals of size $\frac{1}{N^{k+2}}$ that can be destroyed when passing from $X_{\frac{1}{N^{k+1}}}$ to $X_{\frac{1}{N^{k+2}}}$, bounded by $9N\log_2 N$.
• Apply the pigeonhole principle to show that at least one block of $N$ consecutive intervals in the $k$-good structure survives the transition, ensuring $k+1$-goodness.

Experimental results

Research questions

• RQ1 What is the maximal length of an arithmetic progression that can be embedded in the middle $\frac{1}{N}^\text{th}$ Cantor set?
• RQ2 Is the bound of $\frac{N}{100\log_2 N}$ on the length of such progressions tight, or can it be improved?
• RQ3 Can the result be extended to configurations beyond arithmetic progressions, such as more general patterns or bilipschitz images?
• RQ4 Does there exist an $N$ such that every bilipschitz image of the middle $\frac{1}{N}^\text{th}$ Cantor set contains a 3-term arithmetic progression?
• RQ5 How does the structure of the Cantor set influence the existence of long arithmetic progressions under translation and scaling?

Key findings

• The paper proves that for any $N$, the middle $\frac{1}{N}^\text{th}$ Cantor set contains arithmetic progressions of length $\frac{N}{100\log_2 N}$, which grows with $N$.
• The existence of such progressions is established via induction on $k$-good intervals, showing that non-empty intersections of translated Cantor sets persist at all levels.
• The number of intervals destroyed when refining from $X_{\frac{1}{N^{k+1}}}$ to $X_{\frac{1}{N^{k+2}}}$ is bounded by $9N\log_2 N$, ensuring sufficient survival of intervals to maintain $k+1$-goodness.
• The bound on the number of intervals intersected by a given interval $J$ of size $\frac{1}{N^k}$ is at most $2^{L-k-1}$, which is maximized when $J$ lies within the residual set at level $k$.
• The construction implies that configurations of size proportional to $\frac{N}{\log N}$ exist in the Cantor set, and the method is robust under bilipschitz maps with controlled constants.
• The result is consistent with and later confirmed by Broderick, Fishman, and Simmons using Schmidt’s game, validating the bound’s correctness and optimality up to constants.