On the number of permutation-twisted dot products

Abstract: For distinct real numbers $a_1, \ldots, a_n$ and distinct real numbers $b_1, \ldots, b_n$, consider the sum $S=\sum_{i=1}ⁿ a_i b_{π(i)}$ as $π$ ranges over the permutations of $[n]$. We show that this sum always assumes at least $Ω(n<sup>3)$ distinct values, which is optimal. This ``support'' bound complements recent work of Do, Nguyen, Phan, Tran, and Vu on the anticoncentration properties of $S$ when $π$ is chosen uniformly at random.

• The paper proves that two real n-tuples with distinct entries generate at least c n³ distinct permutation-twisted dot products for an absolute constant c>0, matching the Θ(n³) order of growth.
• The authors reduce the problem to subset sums by selecting Ω(n) disjoint transpositions whose paired differences form superadditive sequences, then establish a cubic subset-sum lower bound through scale separation and recursion.
• The method extends to products of d superadditive sequences with Ω(m^{d+1}) subset sums, motivating an open conjecture that d-fold permutation-twisted products attain Ω(n^{d+1}) distinct values.

Overview

This paper, by Carpenter, Defant, and Kravitz (2601.15276), resolves a natural "support" question in the anticoncentration theory of permutation-twisted dot products. Given two $n$-tuples $a,b\in\mathbb{R}^n$ with distinct entries, the authors study the set

$$\mathcal{S}(a,b)=\left\{\sum_{i=1}^n a_i b_{\pi(i)} : \pi\in\mathfrak{S}_n\right\}$$

of all values attained by the dot product as $\pi$ ranges over the symmetric group. Their main theorem states that there is an absolute constant $c>0$ such that $|\mathcal{S}(a,b)|\geq c n^3$ for all such $a,b$. The bound is sharp up to constants: when $a=b=(1,2,\ldots,n)$, one has $|\mathcal{S}(a,b)|=(1/6+o(1))n^3$.

The problem is motivated by recent work of Do, Nguyen, Phan, Tran, and Vu (Do et al., 25 Dec 2025), who proved that for $\pi$ uniform on $a,b\in\mathbb{R}^n$0, each value of $a,b\in\mathbb{R}^n$1 is attained with probability at most $a,b\in\mathbb{R}^n$2 — optimal up to the logarithm — answering a question of Pawlowski arising from Littlewood–Offord theory. The present result complements that work by bounding the size of the support itself; subsequent activity includes Berger–Berkowitz–Devlin–Vu (Berger et al., 7 Jan 2026), Hunter–Pohoata–Zhu (Hunter et al., 9 Jan 2026), and an independent proof of the same cubic bound by Pohoata (Pohoata, 18 Jan 2026) via more complicated methods.

Some distinctness hypothesis is clearly necessary: if either tuple has repeated entries, the sum can be constant over all $a,b\in\mathbb{R}^n$3. The trivial lower bound obtainable by adjacent transpositions from the increasing to the decreasing order of $a,b\in\mathbb{R}^n$4 is only $a,b\in\mathbb{R}^n$5, so the jump to $a,b\in\mathbb{R}^n$6 requires capturing behavior at many scales simultaneously.

Proof strategy: involutions and superadditive sequences

The proof exploits the fact that for a product of disjoint transpositions $a,b\in\mathbb{R}^n$7,

$a,b\in\mathbb{R}^n$8

where $a,b\in\mathbb{R}^n$9. Hence it suffices to find disjoint pairs whose associated products $$\mathcal{S}(a,b)=\left\{\sum_{i=1}^n a_i b_{\pi(i)} : \pi\in\mathfrak{S}_n\right\}$$0 have $$\mathcal{S}(a,b)=\left\{\sum_{i=1}^n a_i b_{\pi(i)} : \pi\in\mathfrak{S}_n\right\}$$1 distinct subset sums. This reduces the permutation problem to a purely additive-combinatorial one about subset sums.

Greedy pair selection. After sorting $$\mathcal{S}(a,b)=\left\{\sum_{i=1}^n a_i b_{\pi(i)} : \pi\in\mathfrak{S}_n\right\}$$2 increasingly, the authors iteratively select a pair $$\mathcal{S}(a,b)=\left\{\sum_{i=1}^n a_i b_{\pi(i)} : \pi\in\mathfrak{S}_n\right\}$$3 among unpaired indices minimizing $$\mathcal{S}(a,b)=\left\{\sum_{i=1}^n a_i b_{\pi(i)} : \pi\in\mathfrak{S}_n\right\}$$4 subject to the constraint that exactly $$\mathcal{S}(a,b)=\left\{\sum_{i=1}^n a_i b_{\pi(i)} : \pi\in\mathfrak{S}_n\right\}$$5 unpaired indices lie strictly between $$\mathcal{S}(a,b)=\left\{\sum_{i=1}^n a_i b_{\pi(i)} : \pi\in\mathfrak{S}_n\right\}$$6 and $$\mathcal{S}(a,b)=\left\{\sum_{i=1}^n a_i b_{\pi(i)} : \pi\in\mathfrak{S}_n\right\}$$7. The procedure halts only when no admissible pair remains, which forces $$\mathcal{S}(a,b)=\left\{\sum_{i=1}^n a_i b_{\pi(i)} : \pi\in\mathfrak{S}_n\right\}$$8; thus it always produces a linear number of pairs. Writing $$\mathcal{S}(a,b)=\left\{\sum_{i=1}^n a_i b_{\pi(i)} : \pi\in\mathfrak{S}_n\right\}$$9, the crucial property is superadditivity: $\pi$0 whenever $\pi$1. The greedy minimality guarantees this, since the indices between $\pi$2 and $\pi$3 are available at earlier stages and provide witnesses for the bounds $\pi$4 and $\pi$5. Applying the same construction to $\pi$6 yields a second superadditive sequence $\pi$7 with $\pi$8.

Subset-sum lemma. The technical core is that for any two superadditive sequences of positive reals, the set $\pi$9 has at least $c>0$0 distinct subset sums for some absolute $c>0$1. The proof establishes a "scale-separation" claim: there exists $c>0$2 such that $c>0$3. The heuristic is that fractional applications of superadditivity morally give $c>0$4; to make this rigorous while keeping superadditivity "in the integers," the authors partition the multiset consisting of $c>0$5 copies of $c>0$6 (for $c>0$7) into subsequences each of sum between $c>0$8 and $c>0$9, where $|\mathcal{S}(a,b)|\geq c n^3$0, and apply superadditivity to each part. Choosing $|\mathcal{S}(a,b)|\geq c n^3$1 large enough makes the accumulated error factors absorb into the slack.

The claim feeds a recurrence $|\mathcal{S}(a,b)|\geq c n^3$2 for $|\mathcal{S}(a,b)|\geq c n^3$3, the minimum number of distinct subset sums. Setting $|\mathcal{S}(a,b)|\geq c n^3$4, one obtains $|\mathcal{S}(a,b)|\geq c n^3$5. Since iterating $|\mathcal{S}(a,b)|\geq c n^3$6 decreases the argument doubly-exponentially fast, the product of the exponential decay factors converges, so $|\mathcal{S}(a,b)|\geq c n^3$7. Combining the lemma with the extracted sequences (taking $|\mathcal{S}(a,b)|\geq c n^3$8) completes the proof of the cubic lower bound.

Higher-dimensional generalization and a conjecture

The subset-sum lemma generalizes: for any $|\mathcal{S}(a,b)|\geq c n^3$9 superadditive sequences of positive reals, the elementwise products $a,b$0 admit at least $a,b$1 distinct subset sums. The authors note, however, that they do not know whether this admits an interpretation for $a,b$2-fold permutation-twisted products

$a,b$3

and they conjecture that such sets have $a,b$4 distinct values when each tuple has distinct coordinates. This conjecture is stated without proof and remains open.

Limitations and open questions

Several caveats bear directly on the strength of the results. First, the theorem gives only the existence of an unspecified absolute constant $a,b$5; the matching upper-bound example shows the truth lies between $a,b$6 and $a,b$7 per unit of $a,b$8, but the paper does not attempt to optimize $a,b$9. Second, the reduction through disjoint transpositions means the $a=b=(1,2,\ldots,n)$0 distinct sums are witnessed by a restricted family of permutations (involutions times a fixed base permutation); whether the full orbit structure could yield stronger statements is not explored. Third, the higher-dimensional generalization of the subset-sum lemma lacks a known connection to genuine multi-permutation dot-product problems, and the resulting $a=b=(1,2,\ldots,n)$1 conjecture for $a=b=(1,2,\ldots,n)$2 is unverified. Finally, the relationship between this support bound and the anticoncentration estimates of Do et al. — e.g., whether the logarithmic gap in their $a=b=(1,2,\ldots,n)$3 bound can be removed using support-type information — is not addressed here.

Conclusion

The paper establishes an optimal-order lower bound of $a=b=(1,2,\ldots,n)$4 on the number of distinct permutation-twisted dot products of two real $a=b=(1,2,\ldots,n)$5-tuples with distinct entries, via an elementary argument combining a greedy pairing scheme producing superadditive difference sequences with a recursive subset-sum analysis. The method's extension to products of $a=b=(1,2,\ldots,n)$6 superadditive sequences suggests a natural higher-dimensional conjecture whose combinatorial underpinnings remain to be connected to the permutation setting.