Cauchy-Davenport Theorem for abelian groups and diagonal congruences

Cochrane, Todd; Ostergaard, Misty; Spencer, Craig

doi:10.1090/proc/14504

Cauchy-Davenport Theorem for abelian groups and diagonal congruences
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by Todd Cochrane, Misty Ostergaard and Craig Spencer PDF

Proc. Amer. Math. Soc. 147 (2019), 3339-3345 Request permission

Abstract:

We prove an analogue of the Cauchy-Davenport Theorem and Chowla’s Theorem for sum sets in a general abelian group and give an application to diagonal congruences, establishing a best possible estimate for the distribution of solutions of a diagonal congruence $\sum _{i=1}^n a_ix_i^k \equiv c \pmod q$ with an arbitrary modulus.

References

R. C. Baker, Small solutions of congruences, Mathematika 30 (1983), no. 2, 164–188 (1984). MR 737175, DOI 10.1112/S0025579300010512
B. J. Birch, Small zeros of diagonal forms of odd degree in many variables, Proc. London Math. Soc. (3) 21 (1970), 12–18. MR 266855, DOI 10.1112/plms/s3-21.1.12
A. Cauchy, Recherches sur les nombres, J. Ecole Polytechnique 9 (1813), 99–116.
I. Chowla, A theorem on the addition of residue classes: Application to the number $\Gamma (k)$ in Waring’s problem, Proc. Indian Acad. Sci. Section A 1 (1935), 242–243.
Todd Cochrane, Misty Ostergaard, and Craig Spencer, Small solutions of diagonal congruences, Funct. Approx. Comment. Math. 56 (2017), no. 1, 39–48. MR 3629009, DOI 10.7169/facm/1587
Todd Cochrane, Misty Ostergaard, and Craig Spencer, Solutions to diagonal congruences with variables restricted to a box, Mathematika 64 (2018), no. 2, 430–444. MR 3798606, DOI 10.1112/S0025579318000025
H. Davenport, On the addition of residue classes, J. Lond. Math. Soc. 10 (1935), 30–32.
Y. O. Hamidoune and Ö. J. Rödseth, On bases for $\sigma$-finite groups, Math. Scand. 78 (1996), no. 2, 246–254. MR 1414651, DOI 10.7146/math.scand.a-12586
Martin Kneser, Abschätzung der asymptotischen Dichte von Summenmengen, Math. Z. 58 (1953), 459–484 (German). MR 56632, DOI 10.1007/BF01174162
Melvyn B. Nathanson, Additive number theory, Graduate Texts in Mathematics, vol. 164, Springer-Verlag, New York, 1996. The classical bases. MR 1395371, DOI 10.1007/978-1-4757-3845-2
John E. Olson, Sums of sets of group elements, Acta Arith. 28 (1975/76), no. 2, 147–156. MR 382215, DOI 10.4064/aa-28-2-147-156
Misty Ostergaard, Solutions of diagonal congruences with variables restricted to a box, ProQuest LLC, Ann Arbor, MI, 2016. Thesis (Ph.D.)–Kansas State University. MR 3597685
Jane Pitman and D. Ridout, Diagonal cubic equations and inequalities, Proc. Roy. Soc. London Ser. A 297 (1967), 476–502. MR 215785, DOI 10.1098/rspa.1967.0082
Wolfgang M. Schmidt, Small zeros of additive forms in many variables. II, Acta Math. 143 (1979), no. 3-4, 219–232. MR 549776, DOI 10.1007/BF02392094