Additive relations in irrational powers

Technical Summary: Additive Relations in Irrational Powers

Problem Statement
This paper investigates the additive structure of sets of the form $A[c] = \{a^c : a \in A\}$, where $A$ is a finite set of non-negative real numbers and $c$ is a real irrational exponent. The primary focus is on the cardinality of the $k$-fold sumset $kA[c]$ and the $k$-fold additive energy $E_k(A[c])$. Specifically, the author seeks to determine the asymptotic behavior of these quantities when $A$ is a subset of an arithmetic progression and $c$ is irrational, contrasting this with the well-studied case where $c$ is an integer.

Methodology
The paper employs a synthesis of tools from model theory, transcendental number theory, and additive combinatorics:

1. O-minimal Geometry and Point Counting: The core analytic tool is the Pila–Wilkie counting theorem and its recent refinement by Binyamini, Novikov, and Zak (BNZ). The author defines the solution sets to additive equations involving irrational powers as sets definable in the o-minimal structure $\mathbb{R}_{\exp}$. The strategy involves counting rational points of bounded height on these sets. A key innovation utilized is the BNZ result, which improves the error term in the Pila–Wilkie theorem from $O(N^\epsilon)$ to a power of $\log N$ for sets definable in $\mathbb{R}_{\exp}$.
2. Functional Transcendence: To handle the "algebraic part" of the definable sets (semi-algebraic curves), the paper proves a functional transcendence theorem (Theorem 3.5). This theorem establishes that if a continuous group homomorphism $\phi: \mathbb{R}_{>0}^n \to \mathbb{R}_{>0}^n$ does not restrict to a morphism of algebraic groups, the Zariski closure of the image of an irreducible semi-algebraic set is a translate of a connected algebraic subgroup. This result relies on Ax's theorem for the exponential function.
3. Diophantine Approximation and Linear Forms in Logarithms: For the second part of the paper, the author utilizes Baker's theory of linear forms in logarithms and Fel'dman's effective bounds. This is used to establish conditions under which linear combinations of powers of multiplicatively independent integers do not vanish.

Key Contributions and Results

• Expansion of Sumsets for Irrational Powers (Theorem 1.1):
  The main result establishes that for a finite set $A$ contained in an $N$-term arithmetic progression with $|A| \ge (\log N)^{C_1}$, and for any real irrational $c$, the cardinality of the $k$-fold sumset satisfies:
  $$|kA[c]| \sim_k \frac{|A|^k}{k!}$$
  as $|A| \to \infty$. This indicates that the set $A[c]$ behaves like a "generic" set with no non-trivial additive relations, provided the exponent is irrational. The result is uniform in $c$.

• Additive Energy Bounds (Theorem 1.3):
  The expansion result is deduced from an asymptotic formula for the additive energy. The paper proves that the number of solutions to the equation $\sum_{i=1}^s a_i^c = \sum_{j=1}^r b_j^c$ (where the tuples are not permutations of each other) is bounded by $O_s(|A|^\theta (\log N)^{C_2})$, where $\theta = \max(1, \min(r, s-1))$.

  • For the case $r=s=2$, this yields $E_2(A[c]) = 2|A|^2 + O(|A|(\log N)^{C_2})$.
  • The paper demonstrates that this bound is optimal up to a power of $\log N$ by constructing examples where non-trivial solutions exist for specific irrational $c$ derived from geometric progressions.

• Corollary for All Exponents (Corollary 1.2):
  By combining the irrational case with existing results for integer exponents (Hooley, Skinner–Wooley, Salberger, etc.) and a linear independence result by Carr and O'Sullivan, the paper proves that for $A = \{1, \dots, N\}$ and any $c \in \mathbb{R} \setminus \{0, 1, 2\}$:
  $$|A[c] + A[c]| \sim \frac{N^2}{2}$$
  This unifies the behavior of sumsets across rational and irrational exponents, excluding the trivial cases $c=0, 1$ and the quadratic case $c=2$ (where Landau's theorem gives a smaller order).

• Non-vanishing for Well-Approximated Exponents (Theorem 1.4 / 5.1):
  The paper provides a Diophantine approximation criterion ensuring that linear forms in $c$-th powers of multiplicatively independent integers do not vanish. Specifically, if $c$ is sufficiently well-approximated by a rational $a/q$ (within a computable bound $\psi$), then $F(n) = \sum a_i n_i^c \neq 0$ for multiplicatively independent $n_i$.

  • This leads to the conclusion that for any set $S$ of multiplicatively independent positive integers, there exist infinitely many effectively computable real numbers $c$ such that $S[c]$ is linearly independent over $\mathbb{Q}$. This offers a new proof of a fact previously established by Bays–Kirby–Wilkie and Jones–Servi.

Significance and Claims
The paper claims to provide the first uniform lower bounds on the size of sumsets $kA[c]$ for irrational exponents, demonstrating that irrational powers destroy additive structure in a way that is quantifiable and asymptotically maximal. The significance lies in:

1. Uniformity: The results hold uniformly for all irrational $c$, unlike previous works that might depend on specific properties of the exponent.
2. Methodological Innovation: The application of the BNZ refinement of the Pila–Wilkie theorem to problems involving irrational powers allows for a power-saving error term (in terms of $\log N$) that was previously out of reach for such problems.
3. Connection to Transcendence: The work bridges the gap between additive combinatorics and functional transcendence, showing how the non-algebraic nature of the map $x \mapsto x^c$ (for irrational $c$) forces the solution sets of additive equations to be "transcendental" in a way that limits their cardinality.
4. Optimality: The paper explicitly constructs examples showing that the derived bounds are sharp up to logarithmic factors, distinguishing the behavior of irrational powers from the "sum-product" phenomenon which does not yield results of this quality for general sets.

The author notes that the method is not expected to generalize to arbitrary generalized arithmetic progressions (GAPs) without further development in o-minimal point counting with explicit dimension dependence, and that the results for sparser sets (e.g., cardinality $\log \log N$) would require significant improvements to current counting theorems.