Comments on: Sums, differences, and surprises http://bit-player.org/2006/sums-differences-and-surprises An amateur's outlook on computation and mathematics. Fri, 30 Jul 2010 22:41:46 +0000 http://wordpress.org/?v=2.6.3 By: Barry Cipra http://bit-player.org/2006/sums-differences-and-surprises#comment-888 Barry Cipra Sat, 26 Aug 2006 17:04:08 +0000 http://bit-player.org/?p=61#comment-888 " ...I had generated the series by looking at differences of distinct elements, but it turns out that sums of not-necessarily-distinct elements yield the same series. Curious." You say curious, I say obvious. If a-b=c-d, then a+d=b+c. Either all four are distinct, or else two are equal, say a=d. My own decline is apparent in the assertion that {1,2,...,m-1} is the difference set for {0,1,2,...,m}. I meant {1,2,...,m}. As one randomly "grows" a set S from among the numbers 0 to m, the number of duplicates, either sum or difference, with each new number goes from none to all -- that is, when you add the n+1st number, you (usually) get no duplicates when n is small (compared to m) and nothing but duplicates when n is large. In general you get some fraction, k/n. I wonder what the transition looks like. ” …I had generated the series by looking at differences of distinct elements, but it turns out that sums of not-necessarily-distinct elements yield the same series. Curious.”

You say curious, I say obvious. If a-b=c-d, then a+d=b+c. Either all four are distinct, or else two are equal, say a=d.

My own decline is apparent in the assertion that {1,2,…,m-1} is the difference set for {0,1,2,…,m}. I meant {1,2,…,m}.

As one randomly “grows” a set S from among the numbers 0 to m, the number of duplicates, either sum or difference, with each new number goes from none to all — that is, when you add the n+1st number, you (usually) get no duplicates when n is small (compared to m) and nothing but duplicates when n is large. In general you get some fraction, k/n. I wonder what the transition looks like.

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