Functiones et Approximatio Commentarii Mathematici

Small solutions of diagonal congruences

Todd Cochrane, Misty Ostergaard, and Craig Spencer

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We prove that for $k \geq 2$, $0 <\varepsilon< \frac 1{k(k-1)}$, $n>\frac {k-1}{\varepsilon }$, prime $p> P(\varepsilon, k)$, and integers $c,a_i$, with $p \nmid a_i$, $1 \le i \le n$, there exists a solution $\underline{x}$ to the congruence $$ \sum_{i=1}^n a_ix_i^k \equiv c \mod p $$ in any cube $\mathcal{B}$ of side length $b \ge p^{\frac 1k + \varepsilon}$. Various refinements are given for smaller $n$ and for cubes centered at the origin.

Article information

Funct. Approx. Comment. Math., Volume 56, Number 1 (2017), 39-48.

First available in Project Euclid: 27 January 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11D79: Congruences in many variables 11D72: Equations in many variables [See also 11P55]
Secondary: 11L03: Trigonometric and exponential sums, general

diagonal congruences in many variables exponential sums


Cochrane, Todd; Ostergaard, Misty; Spencer, Craig. Small solutions of diagonal congruences. Funct. Approx. Comment. Math. 56 (2017), no. 1, 39--48. doi:10.7169/facm/1587.

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