Bipartite Diophantine Tuples: Bounds & Techniques

• Bipartite Diophantine tuples are defined by two natural number subsets A and B where every product ab shifted by n yields a perfect k-th power, generalizing classical Diophantine m-tuples.
• Sharp quantitative bounds are established using methods like the gap principle and sieve techniques, revealing logarithmic limits on minimal set sizes and power-saving estimates on product sizes.
• The framework extends to multipartite settings, offering insights into conditional results under the ABC conjecture and posing open problems in arithmetic geometry.

A bipartite Diophantine tuple with property $BD_k(n)$ consists of two finite subsets $A, B \subseteq \mathbb{N}$ (with $|A|, |B| \geq 2$), such that for all $a \in A$ and $b \in B$, the shifted product $ab + n$ is a perfect $k$-th power in $\mathbb{N}$. This bipartite notion generalizes classical Diophantine $m$-tuples and, in recent work, provides a unifying framework for bounding the size and structure of various Diophantine-type sets and their shifted higher-power analogues (Tsang et al., 3 Dec 2025, Yip, 2023).

1. Definition and Notation

Let $k \geq 2$ and $n \neq 0$ be fixed integers. Denote $\mathbb{N}$ as the set of positive integers. A pair $(A, B)$ of finite subsets of $\mathbb{N}$, each with cardinality at least 2, is a bipartite Diophantine tuple with property $BD_k(n)$ if

$$ab + n = (x_{a,b})^k \quad \text{for all} \quad a \in A,~b \in B,~x_{a,b} \in \mathbb{N}.$$

In symbols: $$(A,B) \text{ has property } BD_k(n) \iff \forall\,a \in A,\,b \in B:~ab + n \in \{ x^k: x \in \mathbb{N} \}.$$ This structure interpolates between classical Diophantine tuples ($n = 1, A = B = C$) and various generalized settings, including shifted and higher-power cases.

2. Quantitative Bounds and Structural Results

Yip establishes sharp unconditional upper bounds on both the cardinalities and product sizes of bipartite Diophantine tuples.

Let $$BM_k(n) = \sup\{ \min\{|A|,|B|\} : (A,B) \text{ satisfies } BD_k(n) \}$$. As $|n| \to \infty$,

$$BM_k(n) \ll \log|n| \quad\text{and specifically}\quad BM_k(n) \leq \left( \frac{4\phi(k)}{k-2} + o(1) \right)\log|n|,$$

where $\phi$ is Euler’s totient function with an absolute implied constant (Yip, 2023).

For larger minimal block sizes, define $$PM_k(n, R) = \sup \{ |A|\cdot|B| : (A,B) \text{ satisfies } BD_k(n),~\min\{|A|,|B|\} \geq R \}$$. For explicit thresholds $R = r_k$ ($r_3 = 9, r_4 = 6, r_5 = 5, r_k = 4$ for $k \geq 6$): $$PM_k(n,r_k) \ll_{k, \varepsilon} |n|^{t_k/k+\varepsilon},$$ with $t_6 = 29/4$ and, for $k \geq 7$,

$t_k = \frac{k^2 + k - 4}{k^2 - 6k + 6}.$

When $k \geq 6$ and $\min\{|A|,|B|\} \geq 4$, one obtains the power-saving bound $|A|\cdot|B| \ll |n|^{1+\varepsilon}$.

3. Generalizations and Conditional Results

A significant generalization of the Bugeaud–Dujella theorem extends from the classical $n = 1$ case to arbitrary nonzero shifts. If $k \geq 4$ and for $a_1 < a_2 \leq b_1 < \cdots < b_m$, all $a_i b_j + n$ ($i = 1,2$) are perfect $k$-th powers, then for any $\epsilon > 0$,

$$m \ll_{k, \epsilon} |n|^{\frac{\phi(k)}{k(k-3)}+\epsilon}.$$

For $k = 3$, the bound is $m \ll |n|^{34/27 + \epsilon}$ (Tsang et al., 3 Dec 2025).

On a conditional basis (assuming the ABC conjecture), minimal sizes $\ell = \ell(k, n)$ can be determined such that any bipartite $BD_k(n)$ tuple with $|A| = \ell$ has $|B|$ bounded as a function of $k$ and $n$:

$$\ell(k, n) = \begin{cases} 2 & k \geq 6, \ 3 & k = 4,5, \ 5 & k = 3. \end{cases}$$

Explicit power-saving bounds under ABC are established for these regimes, using simultaneous Pell-type equations and inductive bootstrapping with the gap principle.

4. Connections to Other Diophantine Structures

Bipartite Diophantine tuples serve as a central notion linking numerous variants and extensions:

• Strongly $A$-Diophantine Sets (Banks–Luca–Szalay): For a fixed $A \subset \mathbb{N}$, a set $S \subset \mathbb{N}$ is strongly $A$-Diophantine if all shifted subset-products $1 + \prod_{s \in R} s \in A$. The construction reduces questions about large strongly Diophantine sets to bipartite $BD_k(n)$ tuples using partitions of subset products.
• Kihel–Kihel $P_n^{(k)}$-sets: A set $A = \{a_1, \ldots, a_m\}$ is a $P_n^{(k)}$-set if, for every $k$-element subset $J$, $\prod_{j \in J} a_j + n$ is a perfect $k$-th power. Tsang–Yip make bounds explicit by partitioning the $\binom{m}{k}$ products into two blocks to produce a bipartite $BD_k(n)$-tuple. The resulting bound: $|A| \ll_k k + \frac{\log(|n|+1)}{k},$ giving explicit finiteness results for $P_n^{(k)}$-sets.

5. Auxiliary Techniques

The proof strategies employ a range of Diophantine, combinatorial, and sieve-theoretic tools:

• Gap Principle: If $a_1 < a_2$ and $b_1 < b_2$ with all $a_i b_j + n$ perfect $k$-th powers, then $b_2 \gg b_1^{k-1}/|n|^k$; repeated application yields super-exponential growth unless set sizes are small.
• Thue–Siegel/Evertse Bounds: Used to control large solutions to $a_2 x_i^k - a_1 y_i^k = n(a_2 - a_1)$.
• Stepanov’s Method over $\mathbb{F}_p$: Bounding the cardinalities of product sets in shifted multiplicative subgroups modulo primes.
• Gallagher’s Larger Sieve: Ensures that $\min\{|A|,|B|\} = O(\log N)$ for $AB + n \subseteq \{ x^k \}$ within $[1,N]$.

These methods interact to establish both unconditional and conditional bounds on the size and product of the tuples.

6. Examples and Applications

• Classical Diophantine Quadruples: E.g., $\{1, 3, 8, 120\}$ split as $A = \{1, 120\}, B = \{3, 8\}$, as a $BD_2(1)$ pair.
• Hilbert Cubes in Shifted Powers: A multiplicative Hilbert cube $H^\times(a_0; a_1, \dots, a_d)$ contained in $\{x^k - n: x \in \mathbb{N} \}$ yields, after partitioning, a bipartite tuple; the bounds imply $d \ll_k \sqrt{\log(|n|+1)}$ for $k \geq 3$.

An explicit unconditional bound for $k = 2$ (squares) gives $d \leq 132$ when $a_0 = 1$ and $d \leq 9$ for $k \geq 3$.

7. Open Problems and Directions for Further Research

• The general case for $k = 2$ (shifted squares) remains unresolved: it is conjectured that for each $n \neq 0$, there exists an absolute constant $C(n)$ such that any $BD_2(n)$-tuple has $\min\{|A|,|B|\} \leq C(n)$. Confirmed only for small shifts $n = \pm 1$.
• Eliminate dependence on the ABC conjecture, potentially via sharper Thue bounds or uniformity results from arithmetic geometry.
• Refine the exponents in all power-saving bounds.
• Investigate “multipartite” Diophantine tuples involving more than two subsets.
• Analyze analogues over number fields, function fields, and finite fields, where different sum–product behaviors arise.
• Study effective and algorithmic enumeration of large $BD_k(n)$-tuples for fixed small $n$ and $k$.

Bipartite Diophantine tuples thus unify and control multiple classical and recent extensions in the theory of Diophantine tuples, serving as a versatile device in modern research on multiplicative and shifted Diophantine problems (Tsang et al., 3 Dec 2025, Yip, 2023).