Timeline for Number of solutions to linear diophantine equations, with natural coefficients in a box
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Jun 1, 2020 at 4:45 | comment | added | Pat Devlin | There is a lot known on this sort of thing. You’re asking for the “atom probabilities” for a sort of Rademacher sum (see also “anticoncentration”). It has to do with the additive structure of the coefficients $a_i$. It gets into something called “inverse Littlewood-Offord theory” with folks like Terry Tao and Van Vu. | |
| May 28, 2020 at 4:39 | comment | added | Avi Tachna-Fram | Do you happen to know the asymptotics of this function in terms of a_i's, d, k? Not the f function I know that one, but the inclusion exclusion sum you gave. | |
| May 27, 2020 at 23:35 | comment | added | Avi Tachna-Fram | Oh I see thats the number of solutions to the unconstrained problem, that makes sense. Thank you so much! | |
| May 27, 2020 at 7:57 | comment | added | Max Alekseyev | Yes, $A$ can be the empty subset, which corresponds to term $f(c)$. | |
| May 27, 2020 at 5:52 | comment | added | Avi Tachna-Fram | Can you explain please. It seems like this is using inclusion exclusion on d, but it seems like it uses f(c) in the answer since that is the term in the sum for the empty set. | |
| May 26, 2020 at 16:06 | comment | added | Max Alekseyev | Let $k,a_i$ be fixed, and $f(c)$ denotes the number of solution to $\sum_{i=1}^k a_i x_i=c$. Then by inclusion-exclusion the number of solutions bounded by $x_i\leq d$ equals $$\sum_{A\subseteq\{a_1,\dots,a_k\}} (-1)^{|A|} f\big(c-(d+1)\cdot\sum_{a\in A} a\big).$$ | |
| May 26, 2020 at 9:55 | review | First posts | |||
| May 26, 2020 at 10:34 | |||||
| May 26, 2020 at 9:47 | history | asked | Avi Tachna-Fram | CC BY-SA 4.0 |