On the asymmetric additive energy of polynomials

McGrath, Oliver

doi:10.1090/tran/9144

On the asymmetric additive energy of polynomials
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by Oliver McGrath;

Trans. Amer. Math. Soc. 377 (2024), 4895-4930

DOI: https://doi.org/10.1090/tran/9144

Published electronically: May 17, 2024

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Abstract:

We prove a general result concerning the paucity of integer points on a certain family of 4-dimensional affine hypersurfaces. As a consequence, we deduce that integer-valued polynomials have small asymmetric additive energy.

References

E. Bombieri and J. Pila, The number of integral points on arcs and ovals, Duke Math. J. 59 (1989), no. 2, 337–357. MR 1016893, DOI 10.1215/S0012-7094-89-05915-2
Timothy Daniel Browning, The polynomial sieve and equal sums of like polynomials, Int. Math. Res. Not. IMRN 7 (2015), 1987–2019. MR 3335239, DOI 10.1093/imrn/rnt350
Tim Browning, The divisor problem for binary cubic forms, J. Théor. Nombres Bordeaux 23 (2011), no. 3, 579–602 (English, with English and French summaries). MR 2861076, DOI 10.5802/jtnb.778
T. D. Browning, Equal sums of like polynomials, Bull. London Math. Soc. 37 (2005), no. 6, 801–808. MR 2186712, DOI 10.1112/S0024609305004741
T. D. Browning and D. R. Heath-Brown, The density of rational points on non-singular hypersurfaces. I, Bull. London Math. Soc. 38 (2006), no. 3, 401–410. MR 2239034, DOI 10.1112/S0024609305018412
D. R. Heath-Brown, The density of rational points on curves and surfaces, Ann. of Math. (2) 155 (2002), no. 2, 553–595. MR 1906595, DOI 10.2307/3062125
Changhao Chen, Bryce Kerr, James Maynard, and Igor E. Shparlinski, Metric theory of Weyl sums, Math. Ann. 385 (2023), no. 1-2, 309–355. MR 4542717, DOI 10.1007/s00208-021-02352-x
Christopher Hooley, On the numbers that are representable as the sum of two cubes, J. Reine Angew. Math. 314 (1980), 146–173. MR 555910, DOI 10.1515/crll.1980.314.146
C. Hooley, On another sieve method and the numbers that are a sum of two $h$th powers, Proc. London Math. Soc. (3) 43 (1981), no. 1, 73–109. MR 623719, DOI 10.1112/plms/s3-43.1.73
Christopher Hooley, On another sieve method and the numbers that are a sum of two $h$th powers. II, J. Reine Angew. Math. 475 (1996), 55–75. MR 1396726, DOI 10.1515/crll.1996.475.55
C. Hooley, On exponential sums and certain of their applications, Number theory days, 1980 (Exeter, 1980) NATO Adv. Study Inst. Ser. C: Math. Phys. Sci., vol. 91, Cambridge Univ. Press, Cambridge-New York, 1982, pp. 92–122. MR 697259
Oscar Marmon, Sums and differences of four $k$th powers, Monatsh. Math. 164 (2011), no. 1, 55–74. MR 2827172, DOI 10.1007/s00605-010-0248-2
Roger C. Baker, Marc Munsch, and Igor E. Shparlinski, Additive energy and a large sieve inequality for sparse sequences, Mathematika 68 (2022), no. 2, 362–399. MR 4418450, DOI 10.1112/mtk.12140
Serge Lang and André Weil, Number of points of varieties in finite fields, Amer. J. Math. 76 (1954), 819–827. MR 65218, DOI 10.2307/2372655