A paucity problem associated with a shifted integer analogue of the divisor function

Heap, Winston; Sahay, Anurag; Wooley, Trevor D.

doi:10.1016/j.jnt.2022.05.006

Cited by 4 publications

(5 citation statements)

References 3 publications

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“…Related paucity results (for linear 𝑃(𝑥) = 𝑥 − 𝛼 with 𝛼 ∈ ℂ irrational) were obtained independently by Bourgain, Garaev, Konyagin, and Shparlinski [3,Corollary 27] and Heap, Sahay, and Wooley [13, Theorems 1.1 and 1 .2]. By considering certain algebraic norms, it may be possible to use our result (Theorem 1.1) to obtain a version of those results (of [3] and [13]), weaker in the error term. However, we are not aware of any known or easy implication in the other direction.…”

Section: Resultsmentioning

confidence: 61%

Paucity phenomena for polynomial products

Wang,

2024

Bulletin of London Math Soc

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Let be a polynomial with at least two distinct complex roots. We prove that the number of solutions to the equation (for any ) is asymptotically as . This solves a question first proposed and studied by Najnudel. The result can also be interpreted as saying that all even moments of random partial sums match standard complex Gaussian moments as , where is the Steinhaus random multiplicative function.

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Section: Resultsmentioning

confidence: 61%

Paucity phenomena for polynomial products

Wang,

2024

Bulletin of London Math Soc

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“…Let P, d, e P be as above. Then for integers k, N ≥ 1, we have A related paucity result (where one takes P to be a linear polynomial with irrational coefficients) was obtained independently by Bourgain, Garaev, Konyagin, and Shparlinski [2] and Heap, Sahay and Wooley [9]. Though it may be possible (by considering certain norms) to use our result to obtain a weaker version of their result (weaker in the error term), we are not aware of any (known or easy) implication in the other direction.…”

Section: Introductionmentioning

confidence: 86%

Paucity phenomena for polynomial products

Wang¹,

Xu²

2022

Preprint

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Let P (x) ∈ Z[x] be a polynomial with at least two distinct complex roots. We prove that the number of solutions (x 1 , . . . , x k , y 1 , . . . , y k ) ∈ [N ] 2k to the equationThis solves a question first proposed and studied by Najnudel. The result can also be interpreted as saying that all even moments of random partial sums 1 √ N n≤N f (P (n)) match standard complex Gaussian moments as N → +∞, where f is the Steinhaus random multiplicative function.

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“…We see from (2.7) that (ξ + w 1 ) • • • (ξ + w s ) divides Θ. Thus, an elementary divisor function estimate (see, for example, [6, Theorem 317]) shows the number of possible [7] Paucity for symmetric Diophantine equations 7 choices for ξ + w 1 , . .…”

Section: Multiplicative Relations From Symmetric Polynomialsmentioning

confidence: 99%

The Paucity Problem for Certain Symmetric Diophantine Equations

Wooley

2022

Bull. Aust. Math. Soc.

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Let $\varphi _1,\ldots ,\varphi _r\in {\mathbb Z}[z_1,\ldots z_k]$ be integral linear combinations of elementary symmetric polynomials with $\text {deg}(\varphi _j)=k_j\ (1\le j\le r)$ , where $1\le k_1<k_2<\cdots <k_r=k$ . Subject to the condition $k_1+\cdots +k_r\ge \tfrac {1}{2}k(k-~1)+2$ , we show that there is a paucity of nondiagonal solutions to the Diophantine system $\varphi _j({\mathbf x})=\varphi _j({\mathbf y})\ (1\le j\le r)$ .

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A paucity problem associated with a shifted integer analogue of the divisor function

Cited by 4 publications

References 3 publications

Paucity phenomena for polynomial products

Paucity phenomena for polynomial products

Paucity phenomena for polynomial products

The Paucity Problem for Certain Symmetric Diophantine Equations

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