Letting $\Omega(n)$ denote the number of prime factors of $n$ counted with multiplicity, Rivat, Sárközy and Stewart (1999) proved a result regarding maximal cardinalities of sets ${\cal A},{\cal B}\subset\{1,\ldots,N\}$ so that for every $a\in{\cal A}$ and $b\in{\cal B}$, $\Omega(a+b)$ is even. This paper extends their work in several directions. The role of $\lambda(n)=(-1)^{\Omega(n)}$ is generalized to all non-constant completely multiplicative functions $f:\mathbb{N}\rightarrow \{-1,1\}$. Rather than just $\Omega$ being even on ${\cal A}+{\cal B}$, we extend the result to all possible parities of $\Omega$ on ${\cal A}$, ${\cal B}$, and ${\cal A}+{\cal B}$. Furthermore, we prove that many such pairs $({\cal A},{\cal B})$ exist. Results from Ramsey theory and extremal graph theory are used.
Funct. Approx. Comment. Math.
52(2):
263-281
(June 2015).
DOI: 10.7169/facm/2015.52.2.6
Y. Buttkewitz, C. Elsholtz, Patterns and complexity of multiplicative functions, J. London Math. Soc. (2) 84 (2011), 578–594. MR2855791 1253.11006 10.1112/jlms/jdr026 Y. Buttkewitz, C. Elsholtz, Patterns and complexity of multiplicative functions, J. London Math. Soc. (2) 84 (2011), 578–594. MR2855791 1253.11006 10.1112/jlms/jdr026
J. Cassaigne, S. Ferenczi, C. Mauduit, J. Rivat, and A. Sárközy, On finite pseudorandom binary sequences III: The Liouville function, I, Acta Arith 87 (1999), 367–390. MR1671629 0920.11053 J. Cassaigne, S. Ferenczi, C. Mauduit, J. Rivat, and A. Sárközy, On finite pseudorandom binary sequences III: The Liouville function, I, Acta Arith 87 (1999), 367–390. MR1671629 0920.11053
E.S. Croot, On the oscillations of multiplicative functions taking values $\pm 1$, J. Number Theory 98 (2003), 184–194. MR1950444 1090.11062 10.1016/S0022-314X(02)00024-0 E.S. Croot, On the oscillations of multiplicative functions taking values $\pm 1$, J. Number Theory 98 (2003), 184–194. MR1950444 1090.11062 10.1016/S0022-314X(02)00024-0
H. Daboussi and A. Sárközy, On pseudorandom properties of multiplicative functions, Acta Math. Hungar. 98 (2003), 273–300. MR1961098 1026.11063 10.1023/A:1022834227653 H. Daboussi and A. Sárközy, On pseudorandom properties of multiplicative functions, Acta Math. Hungar. 98 (2003), 273–300. MR1961098 1026.11063 10.1023/A:1022834227653
C. Elsholtz, Triples of primes in arithmetic progressions, (English summary), Q. J. Math. 53 (2002), no. 4, 393–395. MR1949150 1045.11060 10.1093/qjmath/53.4.393 C. Elsholtz, Triples of primes in arithmetic progressions, (English summary), Q. J. Math. 53 (2002), no. 4, 393–395. MR1949150 1045.11060 10.1093/qjmath/53.4.393
P. Erdős, On extremal problems of graphs and generalized graphs, Israel J. Math. 2 (1964), 183–190. MR183654 0129.39905 10.1007/BF02759942 P. Erdős, On extremal problems of graphs and generalized graphs, Israel J. Math. 2 (1964), 183–190. MR183654 0129.39905 10.1007/BF02759942
P. Frankl, R.L. Graham, V. Rödl, Quantitative theorems for regular systems of equations, J. Combin. Theory Ser. A 47 (1988), 246–261. MR930955 0654.05002 10.1016/0097-3165(88)90020-9 P. Frankl, R.L. Graham, V. Rödl, Quantitative theorems for regular systems of equations, J. Combin. Theory Ser. A 47 (1988), 246–261. MR930955 0654.05002 10.1016/0097-3165(88)90020-9
Z. Füredi, Turán type problems, in Surveys in Combinatorics, 1991, (ed. A. D. Keedwell), London Math. Soc. Lecture Notes 166, Cambridge University Press, Cambridge (1991), 253–300. Z. Füredi, Turán type problems, in Surveys in Combinatorics, 1991, (ed. A. D. Keedwell), London Math. Soc. Lecture Notes 166, Cambridge University Press, Cambridge (1991), 253–300.
R. Graham, V. Rödl, A. Ruciński, On Schur properties of random subsets of integers, J. Number Theory 61 (1996), 388–408. MR1423060 0880.05081 10.1006/jnth.1996.0155 R. Graham, V. Rödl, A. Ruciński, On Schur properties of random subsets of integers, J. Number Theory 61 (1996), 388–408. MR1423060 0880.05081 10.1006/jnth.1996.0155
K. Gyarmati, On divisibility properties of integers of the form $ab+1$, Period. Math. Hungarica 43 (2001), 71–79. MR1830566 10.1023/A:1015229531017 K. Gyarmati, On divisibility properties of integers of the form $ab+1$, Period. Math. Hungarica 43 (2001), 71–79. MR1830566 10.1023/A:1015229531017
P.A. Parrilo, A. Robertson, D. Saracino, On the asymptotic minimum number of monochromatic 3-term arithmetic progressions, J. Combin. Theory Ser. A 115 (2008), 185–192, MR2378863 1210.05172 10.1016/j.jcta.2007.03.006 P.A. Parrilo, A. Robertson, D. Saracino, On the asymptotic minimum number of monochromatic 3-term arithmetic progressions, J. Combin. Theory Ser. A 115 (2008), 185–192, MR2378863 1210.05172 10.1016/j.jcta.2007.03.006
K.F. Roth, Sur quelques ensembles d'entiers, C.R. Acad. Sci. Paris 234 (1952), 388–390. MR46374 0046.04302 K.F. Roth, Sur quelques ensembles d'entiers, C.R. Acad. Sci. Paris 234 (1952), 388–390. MR46374 0046.04302
K.F. Roth, On certain sets of integers, J. London Math. Soc. 28 (1953), 104–109. MR51853 10.1112/jlms/s1-28.1.104 K.F. Roth, On certain sets of integers, J. London Math. Soc. 28 (1953), 104–109. MR51853 10.1112/jlms/s1-28.1.104
A. Schinzel, W. Sierpiński, Sur certaines hypothèses concernant les nombres premiers, Acta Arith. 4 (1958), 185–208; Corrigendum: ibid. 5 (1959), 259. MR106202 A. Schinzel, W. Sierpiński, Sur certaines hypothèses concernant les nombres premiers, Acta Arith. 4 (1958), 185–208; Corrigendum: ibid. 5 (1959), 259. MR106202
G. Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge studies in advanced mathematics 46, Cambridge University Press, 1995. MR1342300 G. Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge studies in advanced mathematics 46, Cambridge University Press, 1995. MR1342300
A. Thomason, On finite Ramsey numbers, European J. Combin. 3 (1982), 263–273. MR679211 0503.05046 10.1016/S0195-6698(82)80038-3 A. Thomason, On finite Ramsey numbers, European J. Combin. 3 (1982), 263–273. MR679211 0503.05046 10.1016/S0195-6698(82)80038-3
P. Varnavides, On certain sets of positive density, J. London Math. Soc. 30 (1959), 358–360. MR106865 10.1112/jlms/s1-34.3.358 P. Varnavides, On certain sets of positive density, J. London Math. Soc. 30 (1959), 358–360. MR106865 10.1112/jlms/s1-34.3.358
E. Wirsing, Das asymptotische Verhalten von Summen über multiplikative Funktionen II, Acta Math. Acad. Sci. Hungar. 18 (1967), 411–467. MR223318 10.1007/BF02280301 E. Wirsing, Das asymptotische Verhalten von Summen über multiplikative Funktionen II, Acta Math. Acad. Sci. Hungar. 18 (1967), 411–467. MR223318 10.1007/BF02280301