Sidon sets with $\Delta$-separated sumsets in additive number theory

Technical Summary: Sidon Sets with $\Delta$-Separated Sumsets in Additive Number Theory

1. Problem Statement and Definitions
This paper addresses the problem of constructing and bounding the size of subsets of integers with specific separation properties in their sumsets. Let $A$ be a nonempty set of integers and $\Delta$ a positive integer. The set $A$ is defined as $\Delta$-separated if $|a - a'| \ge \Delta$ for all distinct $a, a' \in A$.

The paper generalizes the classical concept of $B_h$-sets (where every element in the $h$-fold sumset $hA$ has a unique representation). A set $A$ is a $B_h$-set if $r_{A,h}(n) \le 1$ for all $n$, where $r_{A,h}(n)$ counts the number of $h$-tuples summing to $n$. A $B_{h,\Delta}$-set is a $B_h$-set whose sumset $hA$ is $\Delta$-separated. Specifically, for a $B_{2,\Delta}$-set (a Sidon set with $\Delta$-separated sumset), the condition is that for any $a, b, c, d \in A$ with $\{a, b\} \neq \{c, d\}$, we have $|(a+b) - (c+d)| \ge \Delta$.

The primary objective is to determine $F_{h,\Delta}(n)$, the cardinality of the largest $B_{h,\Delta}$-set contained in the integer interval $\{1, 2, \dots, n\}$. The paper seeks to extend known results for classical $B_h$-sets (where $\Delta=1$) to the $\Delta$-separated case and to compute or bound $F_{h,\Delta}(n)$.

2. Methodology
The paper employs a combination of constructive methods and combinatorial inequalities:

• Dilation Construction: The author utilizes the $\Delta$-dilation of a set, defined as $\Delta^* A = \{ \Delta a : a \in A \}$. Lemma 2 establishes that $\Delta^* A$ is a $B_{h,\Delta}$-set if and only if $A$ is a $B_h$-set. This allows the translation of lower bounds from classical $B_h$-sets to $B_{h,\Delta}$-sets.
• Prime Number Theory: To establish lower bounds, the paper relies on constructions of $B_h$-sets by Bose, Bose-Chowla, Chowla, and Singer, which exist for prime powers $q$. These are combined with Runbo Li's results on the distribution of primes in short intervals ($m - m^\theta < q < m$ with $\theta = 0.52$) to ensure the existence of suitable sets within the interval $\{1, \dots, n\}$.
• Combinatorial Counting and Inequalities: For upper bounds, the paper adapts the Erdős-Turán argument. It utilizes the Cauchy-Schwarz inequality on the counting of pairs within sliding intervals $I_{u,m}$. A key step involves Lemma 4, which proves that a set $A$ is a $B_{2,\Delta}$-set if and only if it possesses a "$\Delta$-separated unique difference set." This equivalence allows the author to bound the number of distinct differences in the set.

3. Key Contributions and Results

• Lower Bound for $F_{h,\Delta}(n)$:
  Theorem 1 provides a lower bound for the size of the largest $B_{h,\Delta}$-set in $\{1, \dots, n\}$. For $h \ge 2$ and $\delta = 0.48$, for all sufficiently large $n$:
  $$F_{h,\Delta}(n) > \left(\frac{n}{\Delta}\right)^{1/h} + O\left(n^{(1/h) - \delta}\right) + O(1)$$
  Corollary 1 specializes this to $B_{2,\Delta}$-sets (Sidon sets), showing $|A'| > (n/\Delta)^{1/2} + O(n^{0.02})$.

• Upper Bound for $B_{2,\Delta}$-sets:
  Theorem 2 establishes an upper bound for the size of a $B_{2,\Delta}$-set contained in $\{1, \dots, n\}$ (where $n \ge \Delta + 2$):
  $$|A| < \left(\frac{2n}{\Delta}\right)^{1/2} + O(n^{1/4})$$
  This result extends the classical Erdős-Turán bound ($|A| \le n^{1/2} + O(n^{1/4})$) to the $\Delta$-separated case, though with a leading constant of $\sqrt{2}$ rather than $1$.

• Asymptotic Limits:
  Corollary 2 derives the asymptotic behavior of the ratio between the set size and the interval scaling:
  $$1 \le \liminf_{n \to \infty} \frac{F_{2,\Delta}(n)}{\sqrt{n/\Delta}} \le \limsup_{n \to \infty} \frac{F_{2,\Delta}(n)}{\sqrt{n/\Delta}} \le \sqrt{2}$$
  The paper notes that the limit is unknown even in the classical case ($\Delta=1$).

• Refinement of the Upper Bound Constant:
  In the "Note" section, the author acknowledges that while the paper's proof yields a constant of $\sqrt{2}$, a subsequent argument by O'Bryant (adapting Linström) proves the stronger inequality $|A| < (n/\Delta)^{1/2} + O(n^{1/4})$. This implies that the limit of the ratio is indeed 1 for all $\Delta \ge 1$.

4. Significance and Open Problems
The paper's significance lies in formally defining and analyzing $B_{h,\Delta}$-sets, bridging the gap between classical additive number theory and sets with separation constraints. It successfully generalizes the construction of large $B_h$-sets to the $\Delta$-separated context and provides the first explicit upper bounds for these sets.

The author identifies several open problems for future research:

1. Extending known results for $B_h[g]$-sets to $B_{h,\Delta}[g]$-sets.
2. Computing the exact value of $F_{h,\Delta}[g](n)$ for all $h \ge 2, g \ge 1$.
3. Determining whether the limit $\lim_{n \to \infty} F_{h,\Delta}[g](n) / (n/\Delta)^{1/h}$ exists.
4. Investigating the number and structural classification of maximum-size $B_{h,\Delta}$-sets within $\{1, \dots, n\}$.

The work remains modest in its claims, presenting bounds and structural lemmas while deferring the resolution of the exact asymptotic limit and classification problems to further research or subsequent literature (as evidenced by the citation of O'Bryant's improvement on the constant).