We prove a theorem which unifies some formulas, for example the counting function, of some sets of numbers including all positive integers,
-free numbers,
-full numbers, etc. We also establish a ...conjecture and give some examples where the conjecture holds.
Sumsets of semiconvex sets Ruzsa, Imre; Solymosi, Jozsef
Canadian mathematical bulletin,
03/2022, Volume:
65, Issue:
1
Journal Article
Peer reviewed
Open access
We investigate additive properties of sets
$A,$
where
$A=\{a_1,a_2,\ldots ,a_k\}$
is a monotone increasing set of real numbers, and the differences of consecutive elements are all distinct. It is ...known that
$|A+B|\geq c|A||B|^{1/2}$
for any finite set of numbers
$B.$
The bound is tight up to the constant multiplier. We give a new proof to this result using bounds on crossing numbers of geometric graphs. We construct examples showing the limits of possible improvements. In particular, we show that there are arbitrarily large sets with different consecutive differences and sub-quadratic sumset size.
When asked to report the 1st digit that comes to mind, a predominant number (28.4%) of 558 persons on the Yale campus chose 7. Three further experiments sought to establish whether this predominance ...was due to an automatic activation process or to a deliberate choice. The 1st experiment with 237 undergraduates showed that the response pattern changed markedly (with 17.3% choosing 7) when the request was for a number between 6 and 15. The 2nd experiment showed that if 7 was mentioned by the E as an example of a response, its frequency dropped significantly (to 16.6%). The 3rd experiment showed that if a number in the 20s was requested the choice pattern remained unchanged (27 was chosen by 27.7%), but if a number in the 70s was requested, 77 was chosen only by 15.5%. All these results are consistent with the idea that Ss choose the response such that it will appear to comply with the request for a spontaneous response.
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