Infinite sumsets in $U^k(\Phi)$-uniform sets

1. Problem Statement and Context

The paper addresses the problem of finding infinite sumset patterns within subsets of the integers ($\mathbb{N}$) that exhibit specific "randomness" properties, defined via Host-Kra local uniformity seminorms ($U_k(\Phi)$).

• Background: Szemerédi's Theorem guarantees arithmetic progressions in sets of positive density. Recent work by Kra, Moreira, Richter, and Roberson extended this to finding infinite sumsets (e.g., $B \cup B^{\oplus 2} \cup \dots \cup B^{\oplus k}$) in sets of positive density.
• The Gap: In general sets of positive density, "parity obstructions" (e.g., the set of odd numbers) often require a shift $t$ in the sumset pattern ($B^{\oplus i} \subset A - t$). The paper investigates whether $U_k(\Phi)$-uniform sets (which are "pseudorandom" up to degree $k$) avoid these obstructions, allowing for sumset patterns without shifts.
• Core Question: If a set $A \subset \mathbb{N}$ is $U_k(\Phi)$-uniform, does it contain rich infinite sumset structures (specifically $B^{\oplus i} \subset A$ for an infinite $B$) without the need for translation?

2. Methodology

The author employs a blend of ergodic theory, topological dynamics, and combinatorial number theory. The methodology follows a "dynamical translation" approach:

1. Furstenberg Correspondence Principle: The problem is translated from the combinatorial setting (subsets of $\mathbb{N}$) to a dynamical setting (measure-preserving systems). Theorem 2.9 and Corollary 2.11 establish that $U_k(\Phi)$-uniform sets in $\mathbb{N}$ correspond to $U_k(X, \mu, T)$-uniform clopen sets in an ergodic system with a generic point.
2. Host-Kra Structure Theory: The analysis relies heavily on the structure of pronilfactors (inverse limits of nilsystems). The $k$-step pronilfactor $Z_k$ characterizes the complexity of the system.
3. Continuous Disintegration: A novel technical tool introduced in Section 3 is the fully supported continuous disintegration. This ensures compatibility between the measure-theoretic fibers and the topological structure of the system, allowing the author to derive topological consequences (like the existence of specific return times) from measure-theoretic properties.
4. Erdős Cubes and Progressions: The paper utilizes the concept of Erdős cubes and Erdős progressions (introduced by Kra et al.) within the dynamical system. These are specific configurations of points in the product space $X^{2^k}$ that correspond to sumset patterns in the return times.
5. Joinings and Nilsystems: The author studies joinings of ergodic systems to analyze intersections of uniform sets with Nil-Bohr$_0$ sets (sets defined by recurrence in nilsystems).

3. Key Contributions and Results

A. Main Theorem on Infinite Sumsets (Theorem 1.1 & 5.1)

The central result establishes that $U_k(\Phi)$-uniform sets contain rich sumset patterns without shifts.

• Statement: Let $k \ge 2$ and $A_1, \dots, A_k \subset \mathbb{N}$ be $U_k(\Phi)$-uniform sets. There exists an infinite set $B \subset \mathbb{N}$ such that:
  $$B^{\oplus i} \subset A_i \quad \text{for all } i = 1, \dots, k$$
  where $B^{\oplus i} = \{b_1 + \dots + b_i \mid b_j \in B, b_1 < \dots < b_i\}$.
• Significance: This removes the need for the shift $t$ required in general positive density sets, proving that the "randomness" of $U_k$-uniformity is sufficient to overcome parity obstructions.

B. Characterization of Regional Proximal Relations (Theorem 1.2 & 4.6)

The paper provides a new characterization of the regional proximal relation of order $k$, denoted $RPk$, in minimal pronilsystems.

• Result: For a minimal pronilsystem $(X, T)$ and points $x, y$, the condition $(x, y) \in RP[k-1](X)$ is equivalent to the existence of infinite sets $B$ such that the sumset pattern $B \cup \dots \cup B^{\oplus k}$ is contained in the return times of $x$ to any neighborhood of $y$.
• Implication: This links the algebraic structure of the system (nilpotency degree) directly to the combinatorial richness of return times.

C. Applications to Substitution Subshifts (Theorem 1.3)

The author applies these results to sequences generated by primitive constant-length substitutions.

• Theorem: If a sequence $\phi(n)$ arising from such a substitution satisfies a specific orthogonality condition against periodic sequences, the set $A = \{n \mid \phi(n) \in E\}$ is $U_\infty(\Phi)$-uniform (uniform for all $k$).
• Specific Examples:
  • Thue-Morse Sequence: The sets $\{n \mid t(n) = a\}$ (for $a=0,1$) are $U_\infty(\Phi)$-uniform. Consequently, they contain the nested sumset pattern $\bigcup_{k \in \mathbb{N}} B_k^{\oplus k}$.
  • Rudin-Shapiro Sequence: Similarly, the sets defined by the Rudin-Shapiro sequence are $U_\infty(\Phi)$-uniform.
• Counterexample: The paper constructs a $U_\infty(\Phi)$-uniform set that is not an IP-set (a set containing all finite sums of some infinite set), refining the understanding of the hierarchy of combinatorial structures.

D. Higher-Order Parity Obstructions (Section 4)

The paper identifies that while $U_k$-uniform sets avoid standard parity obstructions, they are still subject to higher-order obstructions arising from nilsystems. Theorem 4.6 shows that if a system is not a pronilsystem of sufficient degree, these obstructions may persist, preventing the existence of sumsets without shifts.

E. Linear Patterns with Complexity (Theorem 5.11)

The results are generalized to linear patterns defined by vertices of a hypercube with specific complexity. If a family of linear forms has complexity $k-1$, then $U_k(\Phi)$-uniform sets contain the corresponding sumset patterns.

4. Significance and Impact

1. Resolution of Conjectures: The paper provides strong evidence for and partial proofs of Conjecture 3.26 from [55], specifically for the case of $U_k$-uniform sets.
2. Refinement of Pseudorandomness: It clarifies the precise level of "randomness" ($U_k$-uniformity) required to guarantee specific infinite combinatorial structures, distinguishing between sets that merely have positive density and those with structured uniformity.
3. New Tools: The introduction of fully supported continuous disintegration offers a robust method for transferring measure-theoretic results to topological dynamics, particularly for pronilsystems.
4. Concrete Applications: By applying the theory to the Thue-Morse and Rudin-Shapiro sequences, the paper bridges abstract ergodic theory with concrete combinatorial number theory, recovering and extending previous results by Bucci et al. [14].
5. Limitations and Open Questions: The paper explicitly delineates the boundaries of these results, showing via counterexamples (Proposition 6.11) that the theorems cannot be generalized to arbitrary minimal uniquely ergodic systems, and posing open questions regarding the structure of joinings in general ergodic systems.

In summary, Radić's work significantly advances the understanding of infinite sumsets in structured sets, demonstrating that Host-Kra uniformity is a powerful sufficient condition for the existence of complex, shift-free sumset patterns, while simultaneously mapping the precise algebraic obstructions that prevent this in more general dynamical systems.