OFFSET
1,1
COMMENTS
Conjecture 1: Let G be an additive group with |G| > 1, and let p(G) be the minimum of those orders of nonzero elements of G (which is the positive infinity if each nonzero element of G has infinite order). Let n > 1 be an integer, and let A be a finite nonempty subset of G.
(i) For the set n'A = {a_1 + ... + a_n: a_1, ..., a_n are elements of A with a_i not equal to a_{i+1} for all 0 < i < n, and a_n not equal to a_1}, we have |n'A| >= min{p(G), n*|A|-a(n)}.
(ii) For the set n~A = {a_1 + ... + a_n: a_1, ..., a_n are elements of A with a_i not equal to a_{i+1} for all 0 < i < n}, we have |n~A| >= min{p(G), n*|A|-2n+1+(n mod 2)}.
When n > 1 and A = {0,...,k-1}, it is easy to see that |n'A| = n*|A|-a(n) and |n~A| = n*|A|-2n+1+(n mod 2).
Does the following extension of Conjecture 1 also hold?
Conjecture 2: Let G be an additive group with |G| > 1, and let p(G) be as in Conjecture 1. Let A_1, ..., A_n (n > 1) be finite subsets of G with |A_i| > 1 for all i = 1..n.
(i) For the set C = {a_1 + ... + a_n: a_1, ..., a_n are elements of A_1, ..., A_n respectively, with a_i not equal to a_{i+1} for all 0 < i < n, and a_n not equal to a_1}, we have |C| >= min{p(G), |A_1|+...+|A_n|-a(n)}.
(ii) For the set D = {a_1 + ... + a_n: a_1, ..., a_n are elements of A_1, ..., A_n respectively, with a_i not equal to a_{i+1} for all 0 < i < n}, we have |D| >= min{p(G), |A_1|+...+|A_n|-2n+1+(n mod 2)}.
Conjectures 1 and 2 were partially confirmed by Han Wang and Zhi-Wei Sun in their preprint arXiv:2210.12044. - Zhi-Wei Sun, Oct 23 2022
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Paul Balister and Jeffery Paul Wheeler, The Erdos-Heilbronn problem for finite groups, Acta Arith. 140 (2009), 105-118.
Zhi-Wei Sun, Restricted sums of subsets of Z, Acta Arith. 99 (2001), 41-60.
Zhi-Wei Sun and Li-Lu Zhao, Linear extension of the Erdos-Heilbronn conjecture, J. Combin. Theory Ser. A 119 (2012), 364-381.
Van Vu and Terence Tao, Additive Combinatorics, Cambridge Univ. Press, Cambridge, 2006.
Han Wang and Zhi-Wei Sun, On two new kinds of restricted sumsets, arXiv:2210.12044 [math.NT], 2022.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).
FORMULA
a(n) = 2*n+3*(n mod 2)-1. - Chai Wah Wu, Sep 14 2022
From Stefano Spezia, Sep 16 2022: (Start)
If we adopt a(0) = 0, then G.f.: x*(4 - x + x^2)/((1 - x)^2*(1 + x)).
a(n) = a(n-1) + a(n-2) - a(n-3) for n > 3. (End)
Sum_{n>=1} (-1)^n/a(n) = 3*log(2)/4 - Pi/8. - Amiram Eldar, Sep 27 2022
EXAMPLE
a(1) = 2*1 - (-1)*(1+ (1 mod 2)) = 2 + 2 = 4.
a(2) = 2*2 - (-1)^2*(1 + (2 mod 2)) = 4 - 1 = 3.
MAPLE
Suppose a(0) = -1. Then gf := (5*x - 1)/((x^2 - 1)*(x - 1)): ser := series(gf, x, 82):
seq(coeff(ser, x, n), n = 1..80); # Peter Luschny, Sep 17 2022
MATHEMATICA
a[n_]:=a[n]=2n-(-1)^n*(1+Mod[n, 2]);
Table[a[n], {n, 1, 80}]
PROG
(Python)
def A357130(n): return (n<<1)+3*(n&1)-1 # Chai Wah Wu, Sep 14 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Zhi-Wei Sun, Sep 13 2022
STATUS
approved