Authors: and
View More View Less
• 1 Eötvös Loránd Tudományegyetem, Természettudományi Kar, Számítógéptudományi Tanszék Pázmány Péter Sétány 1/C, H-1117 Budapest, Hungary
• 2 Magyar Tudományos Akadémia, Rényi Alfréd Matematikai Kutatóintézet Postafiók 127, H-1364 Budapest, Hungary
Restricted access

USD  \$25.00

### 1 year subscription (Individual Only)

USD  \$800.00

Let A be a set of n real numbers such that the number of distinct twofold sums is a n. We show that the number of twofold products is = c n2/ (a4 log n), and the number of quotients is = c n2/ min (a6, a4 log n) with some absolute constant c. For bounded a this gives the correct order of magnitude for the quotients. For sums we think that the correct order is n2/ (log n)a with some a<1, perhaps with 2 log 2 -1, as a result of Pomerance and Sárközy suggests. We also give more general inequalities for sums, products and quotients formed with different sets. The proofs use geometric tools, mainly the Szemerédi-Trotter inequality.

• Erdös, P. and Szemerédi, E., On sums and products of integers, in: Studies in Pure Math. to the memory of P. Turán, Akadémiai Kiadó (Budapest, 1983), 213-218. MR 86m:11011

Studies in Pure Math. to the memory of P. Turán , () 213 -218.

• Nathanson, M. B., On sums and products of integers Proc. Amer. Math. Soc. 125 (1997), 9-16. MR 97c:11010

'On sums and products of integers ' () 125 Proc. Amer. Math. Soc. : 9 -16.

• Nathanson, M. B. and Tenenbaum, G., Inverse theorems and the number of sums and products, in: Structure theory of set addition, Astérisque 258 (Deshouillers et al., ed.), SMF (Paris, 1999), pp. 195-204. MR 2000h:11110

Structure theory of set addition, Astérisque 258 , () 195 -204.

• Pomerance, C. and Sárközy, A., On products of sequences of integers, in: Coll. Math. Soc. J. Bolyai 51, Number Theory (Budapest 1987), North-Holland Bolyai Társulat (Budapest, 1990), pp. 473-504. MR 91g:11114

Coll. Math. Soc. J. Bolyai 51, Number Theory (Budapest 1987) , () 473 -504.

• Szemerédi, E. and Trotter, W. T. Jr., Extremal problems in discrete geometry, Combinatorica 3 (1983), 381-392. MR 85j:52014

'Extremal problems in discrete geometry ' () 3 Combinatorica : 381 -392.

• Tenenbaum, G., Sur la probabilité qu'un entier possède un diviseur dans un intervalle donné, Compositio Math. 51 (1984), 243-263. MR 86c:11009

'Sur la probabilité qu'un entier possède un diviseur dans un intervalle donné ' () 51 Compositio Math. : 243 -263.

• Elekes, Gy., On the number of sums and products, Acta Arithmetica 81 (1997), 365-367. MR 98h: 11026

'On the number of sums and products ' () 81 Acta Arithmetica : 365 -367.

• Elekes, Gy., Nathanson, M. B. and Ruzsa, I. Z., Convexity and sumsets, J. Number Theory 83 (2000), 194-201. MR 2001e:11020

'Convexity and sumsets ' () 83 J. Number Theory : 194 -201.

• Erdös, P., Some remarks on number theory, Riveon Lematematika 9 (1955), 45-48. MR 17460d

'Some remarks on number theory ' () 9 Riveon Lematematika : 45 -48.

• Erdös, P., An asymptotic inequality in the theory of numbers, Vestnik Leningrad. Univ. 15 no. 13 (1960), 41-49 (in Russian). MR 23#A3720

'An asymptotic inequality in the theory of numbers ' () 15 Vestnik Leningrad. Univ. : 41 -49.

Nov 2020 5 0 0
Dec 2020 1 0 0
Jan 2021 1 0 0
Feb 2021 8 0 0
Mar 2021 0 0 0
Apr 2021 1 1 2
May 2021 0 0 0

## The unit balls of ℒ ( n l ∞ m ) and ℒ s ( n l ∞ m )

Author: Sung Guen Kim

## Another characterization of congruence distributive varieties

Author: Paolo Lipparini

## Lemniscate and exponential starlikeness of regular Coulomb wave functions

Author: İbrahim Aktaş